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0 for i = 1,2, ... On the other hand,
E'-lifn (2, - zm,)= o iniplies 1e
[lSzn6 - Sx,,,ll = 0, which is a contradiction. E
4%
1 .i.
Part 1. Theory of Opcmtor Ideals
Unconditionally Summing Opera,tors
1.7.1. An operator S E B(E, F ) is called unconditionally summing if every weakly swnmable sequence (x,) is mapping into a sequence (Sx,) suminable in the noriii topology. The class of all unconditionally sumriling operators is denoted by U.
Remark. The phrase “unconditionally suinniing” is deduced froni the fact that a,
A Y T ~ ( ,IS ~ ) iiorni convergent for every perinritation
z.
1
1.7.2. Obviously we have the Tht.orem. % i s un operator d e a l . 1.7.3. The following characterization is proved in [ROL, p. 2iOl.
P r o p o s i t i o n . A n operator is E g ( E , F ) i s unconditionally summing if and onhy if the product SX is not an injection for trn?y S E B(co,E).
1.8.
Separahle Operators
1.8.1. An operator S E !$E, F ) ist called sepcinrble if there exist xl, .u2, thai Sxl, Sx2, . . . are driiie in t h r range of S.
... E
E such
The class of all sq)arable operators is denoted by X.
1.8.3. The quickest proof of the following result can be obtained using 2.7.3. T h e o r e m . X is an operrotor i h ! .
1.9.
Kato Operators
1.9.1. We start this section with a Lemiiia. Let S E e ( E ,P) 11~1vet h e property that SJA5is not (in injection for a n y finite codirnemionul subspuce 1%’ of E . ’I’hen,given E --- 0, there exist9 tin infinite dimensional subspace M , such that SJ, ir vppozimnblc rrnd ljSJ,I 5 F , w l m e J, denotes the injection froin MP~ n t oE.
P r o o f . We inductively conrtruct sequences of elemelits x1,xz,. . . E E and functional~a,, f r L , . . . E E satisfyiiig thp following conditions: llzL$= 1 ,
IISx, 2 1
t/32,
l\aklj (= 2”l,
and (x,,uk) = t , l
Since S is not a n injection, we can find 2 , E E such that 1 1 . ~ ~ 1 1 = 1 and IIXxt.,ij5 t i 3 . Moreover, there is (1, E E‘ with ia11j 5 1 and (xl, a,) = 1. Tf sl,..., 5 , E E and n , , . . ., cie E E’ are chosen, piit &In:== (z E E : (x,11,) =
... = (x,a,)
=0).
1. Operator Ideals on Banach Spaces
49
Then M nis finite codimensional, arid SJ$, cannot be an injection. So we may select T , + ~E M , such that jlx,+l[l = 1 and /~S.T,+,]] 5 ~ / 3 ~ +Now ' . there is u:+l E E:' with kill = 1 and ( x ~ +a:,,) ~ , = 1. If n
a,+, :=
-
o \ ak, (xk, an-l/
1
then n
]/an+ll\ 51
+- 12k-1= 2"
and (.zrh+,,a,&,,) = 1.
1
Moreover, ( xi, n,,,)
i 0, there exist S,, S,, ... E a ( E ,P) such that
E
+
E)
jjSj(.
1
g ( E , M i ) and Y , 8(Mi,P) such that Si = Pixi and where M i is a suitable Banach space. P u t M := Z&Wj). m
J i X i and Y := 2 YtQ,. Then
1
1
lpq = IIYll =
(P y
.
IlSjll
Since X E 6 ( E ,M ) and Y E O ( M , F ) , i t follows that S = Y X g W ( E ,F). This proves that 6 & Q2.
3.1.8. The following lemma can be found in [ROL, pp. 264 and 2671. Lemma. For every Bamch space F ,
@(C[O, 11, P ) = m\(C[O, 11, P). We now give a counterexample. P r o p o s i t i o n . The operator ideal B ! .is not aenapoteiif. P r o o f . Let J be the canonical map from C[O, 11 into L,[0, 13. Since &[O, I] is reflexive, we have J E Ill). By the preceding lenima, J i !& Suppose that ?B= ?B2. Then J = Y A , where A E %(C[O, 11, F ) and Y E B(P.&[0, 11). Therefore A E !lB(C[O,11, F ) . Now 3.1.3 implies J = Y-4 E B o = a. This is a contradiction. R e m a r k . By 3.1.6 we have%
+ Op (V).
3.1.9. C o n j e c t u r e . The operator ideal lf is not ideinpotent.
61
3. Products and Quotients o€ Operator Ideals
Quotients of Operator Ideals
3.2.
3.2.1. Let and ‘13 be operator ideals. An operator S 2 ( E , F ) belongs to the lefthand quotient N-l o b if Y S c b(E,Po)for all Y € N(F, Po),where F , is a n arbitrary Banach space. The right-hand quotient 2l o b-l is defined in a n analogous way.
are without any meaning.
Warning. The single syrnbols N-1 and 3.2.2. The following basic statenient is trivial.
Theorem.
a-l o 8 and 21 o b-1are operator ideals.
3.2.3. We next iniprovc 3.1.3. P r o p o s i t i o n . 9) = R o Proof. Let S E g ( E , F ) belong to 5l o m-1. Consider a weak zero sequence (r,). co
and put X ( t , l ):=
E l l x n for (5,)
I , . Then the co-weak conlpactness of U1 and
1
X ( X ’ ) 2 c, imply X E m(ll,E ) . Consequently S X E 5l(Zl,E ) . Sow it follows easily ) is a norm zero sequence. This proves that A o 9l-l G 8. Tlw that ( 8 ~=~(IOXe,) converse inclusion is a conseqrience of 3.1.3.
3.2.4. A proof of the folloiving statement will be given in 28.5.8. P r o p o s i t i o n . A ~ operntor L S E e ( E , F ) belomp to b-I o R i f nnrl sequence (Sx,)with x, E 77, admits a weak Cauchy subsequence.
077&J
if eueiy
R e m a r k . An operator belonging to 9)-1 o fi should be called a lbsenthal operator. R e m a r k . Since the identity map of c, is contained in 8-‘ o a, it follows that
fB*W105l. 3.2.6. P r o p o s i t i o n . A n operator S E B(E, P ) belongs lo 21V109)i f and only if E’-lim x, = o and F”-lim b, = o iwiply lim (Sx,, b,) = 0. n
11
¶
Proof. Let S E E(E, F ) belong to m-1 o B. P u t By := ((y. &)). Then B E E(F, cg). Moreover, M(B”) & c, implies B t m(P,co) by 1.5.4. Consequently B8 E%(E, PO). Now from lim ~~,?3Sx,~l = 0 we obtain lim (Sx,, b,) = 0. n
7k
Conversely, let S B(E, F ) satisfy the condition formulated above. Furthermore suppose that there exists E E m ( F , F,) with BS 4 B(E,F,). Then, given E > 0, we ~ can find a weak zero sequence (x,) such that jlBSxJ 2 E > 0. Choose b,O E U F with (BSXn,bi) = //BSx,/\.By 4.4.7 we have B’ E ?&(Fh, F ‘ ) . Consequently there exists a subsequence (b:,) such that b,, := B’b:< is weakly convergent to some b E l”. Hence Jim (A!!x,,,b,, - b! = 0. This ituplies lim (,?3Sxnl,b:,) = 0 , which is a contradiction. 1
L
R e m a r k . An operator ldonginp to operator.
!&-I
08 should be called a Dwnford-Petti8
3.2.6. We mention without proof a similar statement; cf. [DIC, p. 1791. P r o p o s i t i o n . Anoperntor S implies El’-lim S’b, = 0. U
:Z ( E , 3’) belongs t o X - l o !ID i f a n d o d y i f F-lh b,= n
o
Part 1 . ‘rheorv of Operator Ideals
63
R e m a r k . An operator belonging to X - l o m should be called a Grothendid operutor. 3.2.7. The following theorem can be used to characterize special classes of Banach spaces. T h e o r e m . Let % and B be operator ideals. Then Space (2I-l c d) consists of all Bamch spaces E’ such that N ( E , F ) & b ( E , F ) for arbitrary F .
(a-l
a-l
P r o o f . Tf R E Space o d),then IE E c 6.Hence for every S E have S = SIE E %(E, F ) . The converse implication is likewise trivial.
a(E,P ) we
R e m a r k . The space ideal Space (U o B-l) can be described in an analogous way.
3.2.8. A Banach space is said t o possess the Dunford-Yettis property if it belongs to Space (a-l o 8). 3.2.9. The main examples are treated in [DUN, pp. 494 and 5081. P r o p o s i t i o n 1. Let (9,,u)be any measure ,9pace. Then L l ( 9 ,p) has the Dunford-
Pettis property. P r o p o s i t i o n 2. Let K be aiiy compact Hausdorff space. Then C ( K )has the DunfordPettG property.
3.3.
Notes
The straightforward concept of the product of two operator idesls goes back t o the author, while J. PUHL[i] found the idea of their left- and right-hand quotient; rf. also 31.S. HSIEH[l]. The most examples, given in this chapter, are restatements oi well-knomi facts. The so-called Dunford-Pettis property of Banach spaces was introduced by A. GROTHENDICCH [2]. See also [ L N , pp. 181-184J and [RQL, pp. 263-2661. Recommendations for further rending:
J. DIESTEL [3], D. MORRIS/N. SALINAS [i], W. OOSTENBRIKK [l], N. SALINAS [3], S. SIMONS [Z].
Operator Ideals with Special Properties
4.
This chapter is devoted to constructions of new operator ideals from given ones. Such a rule ne20
: U -7
a""",
(tefining an operator ideal g n e w for every operator ideal a, IS called a procedure. Every procedure carries over itself to space ideals in a canonical way. Considering the class of all stable objects for a given procedure we obtain operator ideals and space ideals having special properties. First the norm closure BrJos is investigated. Then we introduce the radical grad which is of some interest in the spectral theory. The dual operator ideal IUdual consists of all operators S such that S' belongs to N. If 31 = then the operator ideal is called coinpletely symmetric. For exaniple, the classical theorems of 3 . SCHAUDER (1930) and V. GANTMACHER (1940) state that Jt and!& respectively, are completely symmetric. Next the regular hull %reg of an operator ideal is considered. This procedure will be of great importance in the sequel. As A. GROTHENDIECK observed, it may happen that an operator itself does not belong to but becomes a member of if the target Banach space is enlarged. This phenomenon leads to the definition of the injective hull Wl1J.The dual concept of the surjective hull N'"' is obtained in an analogous way. Both constructions were first studied by I. STEPHANI, using Grothendieck's work on tensor products as a basis. Finally, we introduce the maximal hull Urnaxand the niinimal kernel Urn'".These procedures are very important in the context of quasi-normed operator ideals.
a,
a
4.1.
Procedures
4.1.1. A rule
mew: ?il +%new which defines a new operator ideal gnew for every operator ideal 91 i s called a procedurr.
4.1.2. We now list some special properties: qjnew(monotony). (M) If U & b, then (%-)new = anew for all % (idempotence). (I) -4 monotone and idempotent procedure is called a Ivull procedure if U & ?Inewand a kernel procedure if U 2 for all operator ideals, respectively.
anew
4-1.3. Every procedure can be carried over to space ideals by setting Anew..- Space ([Op (A)]"""). R e m a r k . We always have Op (Anew)
[Op (A)]"""
and Space (2P") 2 [Space
but identity does not hold in general.
(a)]""",
64
Pert 1 . Theory of Operator Ideals
4.2.
Closed Operator Ideals
4.2.1. Let U be an operator ideal. An operator S E 2 ( E , P) belougs to the closure UC~OS if there are S,, S,, ... E U ( E ,F ) with lim IIS - Sni/= 0. n
4.2.2. The following statement is evident. Theorem.
CUClos
is an operator ideal.
4.2.3. Moreover, we have the T h e o r e m . The rule
clos: u --f U C l O S is (I hull procedure. 4.2.4. An operator ideal U is called closed if U = CUcloS. 4.2.5. P r o p o s i t i o n . The operator ideals 8 ,$3, m,!B, 3, and X are closed. Proof. By definition we obtain 6 := 810s. The ideals fi and !Xt are treated in
[DUN, pp. 486 and 4831. Let S E !Bclos(E, F), and consider a weal; zero sequence (zn).P u t e := sup Ill:nli. Then, given E > 0, there exists So E @(E,F ) with 1 1 8 - S0li 5 F . We can now find a natural number no such that llSgnll5 E for n 2 no.Consequently llflznll 5 ilsx, - f i o 4 I
+ Il4+Ai
5 (9
+ 1)
F.
Hence S 6 S ( E , F ) . This proves that ‘I) = ( 1 2 ~ 1 ~ . Let S E
UClos(E, P), and
consider a weakly suminable sequence (zn).Put
Then, given E > 0, there exists So E U ( E ,F ) with jjX - Sol/5 natural number no such that
E.
We can select a
Consequently
Hence S E U ( E , F ) . This proves that
u=
3~1~.
... i X(E,F ) with l h I/S- ~ $ 1 1 = 0. n We now choose r,,, x,,, ... E E’ such that Snxnl,SnxnZ,... are dense in iW(Sn).Then, Let S E 3 P s ( E , F ) . Then there are S,, S,,
given 1: E E and F > 0, there exist S, and IIS,z - SnxnJ 5 E. Hence
IlSx
- SZnjll
ynt
satisfying IjS - SnIIllrll 5 F and
5 jjSz - Sllxll + IISnx - S n ~ , i5 l / 28.
4. Operator ldeals with Special Properties
6B
Therefore Sxll, Sq,, Sx,,,... arc dense in M ( S ) , and S is separable. This proves that X = Xclos. R e m a r k . Obviously (fjis the smallest closed operator ideal.
4.2.6. As a n easy consequence of 4.2.5 we obtain the P r o p o s i t i o n . The operator ideuls%-l o R,%-l o'll mid X-I o ZU are closed.
4.2.7. P r o p o s i t i o n . The opeintor ideols 6 nnd
z are closed.
Proof. Let S E W O s ( E , F),and suppose that S J $ is an injection. Tiien there r,xists So E G(E,F ) with /IS- Sol]< j(SJ'&). Now it follows from B.3.11 that S,JfI IS also an injection. Therefore M is finite diniensional. This iiiiplies S < G ( E ,F ) . fIence G = Gclos. The proof of Z = ZClos is completely analogous.
-1.2.8. P r o p o s i t i o n . Let 3 be a n operator ideal. Y ' h m Space
(a)= Space ( a c 1 0 5 ) .
Proof. Let E E Space(aC1os).Since I E E Iuclos, we can find S t a ( E , E ) such that illE- S// < 1. Consequently S = 1, - (I, - S)is invertible. Finally I E = E iinplies E E Space (a).Therefore we have Space (Wlos) Space (a).The converse incluaion is evident. R e m a r k . The procedure clos is the identity for space ideals.
-1.3.
Radical
4.3.1. Let be an operator ideal. An operator S E B(E, F ) belongs to the radical 2 P d if for every L E B(F, E ) there exist U E B(E, E ) and X c U ( E ,E ) such that
U(IE - LS) -1.3.2. T h e o r e m . a
I, - X .
1
n d
is an operator ideal.
Proof. The property (01,) is trivial. (1) Let S,, S, E Urad(E, F ) . Then, given L C B(F, E ) , thare are U , B(E, E ) and XI t %(E,E ) with U l ( I E- LS,) = 1, - X,. We now choose U2 E B(E,E ) and X , E a ( E , E ) such that L',(I, - U,LS,) = I , - X,.Then
U,U,[I, - L(S,
+ S,)] = U,[I1 - X , - U,LS,] = I, - x,
Since X , $. U 2 X , E a(#,E ) , we have S, 4 S,
-
U2Xl.
E Urad(E, P).
(2) Let T E B(E,,E ) , S E Urad(E,F ) , and R E @ ( F , F,). Given L E B(F,, E,), there exist U E B(E, E ) and X E U ( E , E ) with C(IE - Y'LRS) = I E - X . Define the operators U , := I E , + LRSUT and X,:= LRSXT. Clearly So t U(E,,E,) and
U,(IEo- LRSl') = I E ,
LRSl' f I,RSU(I, - 7"LBS)T = Ih, - LRST LRS(I, - X ) T = IEo- Ll2SST = I E , - x,. -
-+
Therefore RSl' 5
t
Pietarh. 0 n r r ; ~ t o r
Ur&d(E,,,Po).
66
P a r t 1. Theory of Operator [deals
4.3.3. T h e o r e m . The rule rad: 'u --f
Wad
i s a hull procedure.
Proof. The property (M) is trivial. If S E % ( E ,F ) and L E B(F, E ) , put U := 1, and X := LS. Then X E % ( E ,E ) and U(IE - L S ) = I , - X . Consequently S E U m d ( E , F). This proves that 'u & W a d . Let S E 2 ( E , F ) belong to (%&)ran. Then, given L 2 ( F , E ) , there are U1 E 2(E,E ) and X , E Umd(E, E ) such that U,(IE - LS) = I , - X,. We can now find U2 E 2(E,E ) and X, E U(E, E ) with U 2 ( I E- X,) = IE - &.. Consequently
U,U,(IE
-
LS) = U,(I,vj - Xi) = IE - X2.
This implies S E Urad. Therefore evident, we have (I).
(Wad)rad
& Wa. Since the converse inclusion is
1.3.4. P r o p o s i t i o n . For every operator ideal 2 l the radical
is closed.
Proof. Let S E 2(E,F ) belong to (Urad)clos. Given L E 2 ( F ,E ) , there is So E Umd(E, F ) such that l!Lll /IS- Soli< 1. We now choose Uo E B(E,E ) and X o c %(E,E ) with
Uo(IE - [IE - L(S - So)]-' LSO) = I E - Xo. Put
u := uo[IE - L(s- so)]-'.
Then
U[IE - LS] = Uo[IE - L(S - So)]-' [I, - L(S - So) - LSo] = Uo(IE - [ I , - L(S - So)]-' LSo) = IE
Consequently S E U m d ( E , F). This proves that
(%md)clos
- Xo.
5 Wad.
R e m a r k. Obviously %cloS & %rad.
4.3.5. P r o p o s i t i o n . Let U be a n operutor ideal. Then a r a d = (Uc1os)rad.
P r o o f . It follows from
U(*los
5 W a d that
(2Ps)rad
E (Urad)rad = U r a d .
R e m a r k . I n 26.7.3 we shall prove that p d = !it&. The operators belonging to this ideal, denoted by ?&are called Gohberg operators (or inessential).
4.3.6. We now improve 4.2.8.
I' r o p o s i t i o 11. Let
be a n operator ideal. Then
Space ( 8 )= Space ( a n d ) . Proof. Let E c Space (Wad). Since I E E Urad, we can find U E 2 ( E , E ) and X E 'u ( E ,E ) such that U ( I , - IJE) = I &- X . Therefore I E = X E 94 and E E Space (a).Hence we have Space (%fad) & Space (a).The converse inclusion is evident. R e m a r k . The procedure rad is the identity for space ideals.
4. Operator Ideals with Special Properties
67
4.3.7. With respect to Problein 2.2.8 i t arises the following C o n j e c t u r e . Let A be a space ideal. Then Op (A)"d is the largest operator ideal % with A = Space (a). 4.3.8. Finally, we show that the asymmetry in the definition of
%pa*
can be removed.
Leniina 1. Let S c U?Zd
U r a d ( l 8 , F ) . Then for every L E B(F, E ) there ex& U E 2(E, E ) X I , X 2 E %(E,E ) such that
U(IE - LS) = 1, - XI and Proof. We can find U E Since
R := I E - U
( I , - LS) U = I B - X,.
B(E,E ) and XI E %(E,E) with C(I, - LS) = IE
=X I
- X,.
- ULS E 2lm*(E, E ) ,
there are Uo E B(E,E ) and Xo E %(E,E ) such that Uo(IE - R)= I E - Xo. We obtain from UoU Xo = I E that
+
+ Xo(IE - LS) U = Uo(IE - XI)U + Xo(IE - LS) U = 18 - X , , X2 := Xo(IE - U + L S U ) + UoXiU E %(E,E). (I, - LS) U
where
=
UoU(IE - LS) U
L e m m a 2. Let S E U r a d ( E , F ) . T h e n for every L E B(F, E ) there exist V E B(P,F ) and Y,, Y , € % ( F ,F ) such that
V ( I F- S L ) = IF - Y, Proof. Apply Lemma 1 to SL E
4.4.
and
( I , - SL) V = IF - Yf.
Wd(F,
F).
Symmetric Operator Ideals
4.4.1. Let U be a n operator ideal. An operator S E &(E,P ) belongs to the dun1 operator ideal %dual if S' c %(F', El). 4.4.2. The following statement is obvious. Theorem.
%dual
is an operator ideal.
4.4.3. Having in mind the property (M) defined in 4.1.2 we formulate the Theorem. T h e rule
dual: % --f
%dual
is rr monotone procedure.
-k%. We now carry over the procedure dual to space ideals. Theorem. Let A be a space ideal. T h e n that E' E A.
Adual
consists of all Banach spaces E such
Proof. By definition E E Adual is equivalent to I E E [Op (A)]dual. This means that = Q J , where J E 2(E',iM),Q E E(M, El), and M E A. In other words, E' is
1,.
68
Part 1. Theory of Operator Ideals
isomorphic to a complemented subspace of each other.
rW.Therefore E’ E A and E
E A d d iniply
5 (Udml. In case % = %dml the operator ideal is said to be cmpletely symmetric. The analogous definitions will be used for space ideals.
4.4.5. An operator idsal U is called symmetric if
4.4.6. The next result is trivial. P r o p o s i t i o n . For every (completely) symmetric oprcttor ideal % the space idecrl Space (a)is (cornpleteZy)symmetric, as well.
4.4.7. We now consider mine examples. P r o p o s i t i o n . T h e operator ideals $, 6, 8, and ?.Bnre completely sFjnimetric. Proof. The formula 5 = $dm1 is evident. In order t o show Q = %dud we need a special technique which will be developed later; cf. 11.7.4. The complete symmetry of 8 and is equivalent to the classical theorems of J. SCRAUDER and V. GAXTMACHER, respectively; [DUN, p. 4851.
4.4.8. P r o p o s i t i o n . l’he operafor ideals 8, U, c x.
x-1
r(nd
X
are not symmetric. iKoreouer,
Proof. The first part of the assertion follows froin the fact that the identity map of 2, belongs to ?B,a, and X, but not to Bdud, Udual. and Xdual. Let S E Xdual(E, F ) . Then there exist bl, bp,... f F’ such that S’b,, S’b,, dense in M(S‘). Choose x,, x2, 5 E’ with IIS‘bJi 5 2 !(xi, S’b,)l and 1ix2]] 5 1.
._.
. .. are
Let 2 1 1 be the subspace spanned by Sx,,Sx,,. .. If Sx 6j M , then there is b E F‘ such b), = 0 for i = 1, 2, ... Since we can find bio with that (Sx,b) $: 0 and ( S X ~ 1 /IS%- X’bioll 5 - IIS’bli, it follows that
4
IIX’bioli 5 2 and therefore
I(z,~,S’biJ1
= 2 ~ ( S Xbil, , ~, b)\
2 2 /jS’bbo- S’b/l
This contradiction iniplies M ( S ) 5 M . Consequently S E X(E,P).
4.4.9. A Banach space E is called quasi-dual if its canonical image is complemented in E”. The class of all quasi-dual Banach spaces is denoted by D. P r o p o s i t i o n . D i s a space ideal. P r o o f . Since the properties (51,) and (51,)are evident, we check (SI,) only. Let E . Then IEo= QJ, where J E L?(E,,E ) and Q 5 C ( E , E,). If E E D, then there exists P E 2(E”,E ) such that I , = PKE. Now IE0= QPKEJ = QPJ“KE,implies E, E D. Therefore (S1,) is satisfied. 4.4.10. An operator S E 2 ( E , P)is said to be dualisable if it belongs to 3 := Op (D). P r o p o s i t i o n . T h e wperntor ideal3 is symmetric, but not completely symmetric.
E,
. WEIS [I?* Proposition. G s ~ = r 21nj = rrZ-I o a. R e m a r k . It seems to be unknown whether there exists an operator ideal 91 such that %inj = G and 9 W = z.
4.8.
Minimal Operator Ideals
i.8.L. We start this section with a general statement. Theorem. Let b be an idempotent operator ideal. '/'hen the rule 2I + 91 o b kerizel procedure.
i.9
a
Proof. We only mention that (3o b) o b = 91 o B2= 2€o b. R e m a r k . The same result holds if we multiply from the left-hand side.
4.8.2. Let 91 be an operator ideal. An operator S E B(E, F ) belongs to the minimal kernel Umln if S = YSoA, where A E 6 ( E ,Eo), So E %(KO. Fo), and Y E (ti(Fo,2'). In other terms, U m i n := 6 o % o 6. 4.8.3. Obviously 3.2.2 yields the T h e o r e m . 2 I m h i s a n operator ideal.
4.8.4. By 3.1.7 and 4.8.1 we have the T h e o r e m . T h e rule
miri: 91 -> ir
?@in
a kernel procedure. R e m a r k . For every space ideal A we get
Amln
= F.
4.3.6. The procedure m2in is trivial for the most operator ideals considered till now. Significant examples will be treated in Chapter 18. P r o p o s i t i o n . Let 91 be
(L
closed opewitor ideal. T h e n W
Proof. The assertion follows froni 6 5 91. 4.8.6. An operator ideal U is called minimal if U = % m k .
i n
= 6.
76
Part 1. Theory of Operator Ideals
4.8.7. From 4.8.5 we immediately obtain the P r o p o s i t i o n . 6 is the only minimal closed opercitor
idtleol.
-1.8.8. The next statement is obvious. P r o p o s i t ion. !!'he operator ideal 8 is ~ n i ? ~ i ~ t i ~ l .
3.9.
Maximal Operator Ideals
4.9.1. 5;1'v start this section with a general StateJll~vlt. Theorem. Let b be cin dernpote~iloperator idml. Then the rule 41 -> % o b-3is a huU procedure. P r o o f . We only shon the idempotence. Let S E %(E,E") belong to (% o 4-l)o %-I. Because of b2= 4,every operator X E 4 ( E 0 ,E ) admits a factorization X = XIXo, where Xo b(E,,El) and X , E B(E,. E ) . Then we have SX1 E o 4-' and S X = (SX,) X , E 3. Therefore o b-1)o b-1& (er o 4-1.The converse inclusion is trivial.
(u
R e m a r k . The saim result holds for the left-hand quoticnt.
4.9.2. Let a hc an operator ideal. An operator 1.5' E E(E, F ) belongs to the maximal huU a m if BXX < a(&,Po)for all X Q(Eo,E ) and B Q ( F ; Po).In other terns. u- := 0-10 'ti 0 6 - 1 .
4.9.3. Obviously 3.2.2 yields the Theorem.
Zma*
is an operatoy ideal.
4.9.4. By 3.1.7 and 4.9.1 we have the T h e o r e m . The rule
m a x : 91 --f a m a x i s a hut1 procedure.
4.9.5. The procedure naax is trivial for the most operator ideals considered till now. Significant examples will be treated in Chapter 17 and 19. P r o p o s i t i o n . Let amax =
a be n cZosed operator
ideal. T h e n
2.
P r o o f . The assertion follows from 6 & 3. 4.9.6. Ar? operator ideal is called nirixininl if rU = Q P'. 4.9.7. From 4.9.5 we iniiiiediately obtain the P r o p o s i t i o n . 2 i s the only rnnximal closed opcivlor ideul. 4.9.8.
At first glance the nest result seems to be somelion, surprising.
P r o p o s i t i o n . !Z'?Le opemior ideal 8 is maximal.
4. Operator Ideals with Special Properties
77
Proof. Let S t 2 ( E , 4') btlonp to 5""".Furtherrl~oresuppose that S is not finite. Using the Hahri-Banach theoreiii we can find X , E i!(Zf, E ) and B, E O ( F , If) such that B,SX, = e,I, and l]Xn]i= ~ ~ =B 1., Hcre ~ ~I , denotrxs the identity inap of Zg, and en is a suitable positive nuiiiber. Form the Hilbert space 11 := Z2(Z;); cf. C.4.1. If
S
:=
- 1
- X,Q,
1
snd B :=
"I
- J,B,,
1 n
n
then X E B(H, E ) and B E Q ( F , H ) . Therefore B S S E 8. This is a contradiction, since Q,BSXJ, = ? r 2 p J n for n = 1, 2 , . ..
4.9.9. We non state the open P r o b l e m . Let A he a space ideal. What about R e m a r k . Obviously
Fmx =
4.9.10. P r o p o s i t i o n . Let (Uw
F.
Wmax
(U Dr an
Amax?
= L, and X m a v = L.
operator ideal. Then
& amax.
Proof. Let S 5 ( U r e g ( E , P). Then, given X E @(Eo,E ) and R E Q ( F , F,), from BSX = BnKFSXit follows that BSX E %(Eo,F0). Hence S E % m d x ( E , F ) .
4.9.11. As an immediate consequence we have the P r o p o s i t i o n . Every mximrrl operator ideal i s regular.
4.9.12. Finally, we deal with the connection between the procedures max and min. P r o p o s i t i o n . Let Iu be an operator ideul. The?z (%min)max = (Umax
a&
((Umax)min = (Umin.
Proof. Let S E 2 ( E , F ) belong to a m a x . We know froiii 3.1.7 that X E Q(E,, E ) and B E B ( F , Po)can be written in th(1 foriii X = S,Xo arid B = BOB,,respectively, X , E @(El,E ) , B, E Q ( F , F , ) , and BoE @(F,, Fo).Now S t where X o E Q(E,,El), implies B,XX, E (U and B S X = Bo(BISX,)X o E (Umln. Thus S i ((Um1n)max. This proves that %ma\ & ((Um1n)mau. The converse inclusion is trivial. Let S E O(E, F ) belong to ((Umax)ml*. Then there are A E Q ( E , ,Yo), So E %""(&,,FO), and Y E @ ( P oF , ) such that S = YSOA.By 3.1.7 we have A = AoA, and Y = YIYo, where A , E W E , El), A , E @(El,Eo), Y o E @(go, F,),and Y , E @(Fl,F ) . Now it that S 2 W n . Consequently follows from S = Y,(YoSoA,)A , and YoS,,AoE (W-)min %mi*. The converse inclusion is trivial.
4.10.
Notes
The concept of a dosed operator ideal is straightforward. The radical of an operator ideal seems t o be defined in this monograp!i for the first time. I n thc special case of compact operators this coiistruction goes back to R. TOOD [l] and I. C . GOHBERQ/A. s. M ~ ~ a r r s /9. I . FELD-
78
Part 1. Theory of Operator Ideals
MAN [l]. See also the lecture notes [CAR, pp. 95-1021. Symmetric and regular as well as minimal and maximal operator ideals were studied by A. PIETSCH [13] and H.-U. SCHWARZ [l], [2]. The concepts of injectivity and snrjectivity are taken from the theory of tensor pro[4]. For further informations we refer t o the work of I. STEPHANI ducts; cf. A. GROTHENDIECK PI, ~31.
Recommendations for further reading:
J. DIESTEL/B.FAIRES[l], V. R. GANTMACHER [l], D. KLEINECKE[l], J. SCHATJDER [l], L. WEIS 111, R. J. U’HITLEY[l], B. Yoon [l].
5.
Closed Operator Ideals on Cla.ssica,lBanaeh Spaces
I n this chapter we investigate the lattice of all closed ideals in the operator algebra of a fixed classical Banach space. The first result of this kind goes back t o J. W. CALKINin 1940. He proved that for l2 there is only one non-trivial closed ideal. This fundamental theorem was extended to the sequence spaces l p , where 1 5 p < 00, and co by I. C. GOHBERG, A. S. MARKUS, and I. A. FELDMAN. Until now no other Banach spaces with this property are known. The situation is completely different in function spaces. Of course, Calkin’s theorem holds for LJO, 11. On the other hand, there are a t least countably many closed ideals in the operator algebra of Lp[O,11, where 1 < p < co and p =# 2. I n the case of C[O, 11 we even construct a n uncountable chain of closed ideals. The problem seems to be unsolved for L,[O, 11. Another generalization of Calkin’s theorem is due to B. GRAMSCHand E. LUFT. Both observed independently that, for a n arbitrary Hilbert space, the lattice of closed ideals is well-ordered.
5.1.
Operators in Sequence Spaces
5.1.1. At this point let E be either l p (1 5 p < co) or co and let F be either 1, (1 (I < 00) or co. The natural projection Pnis defined by pn(t1,
..
En, Sn+l,
-
.) := ( E l ,
* * *,
En,
0, - * * I
and ek denotes the k-th unit sequence. L e m m a 1. Let x l , z2,... E E such that 0 < a-l 5 l ] X k / / 6 < 00 nnd lxil A 1 9 1=o for i 7 k. Then there are X E B(E, E ) and A E B(E,E ) with IlXll f and llAil 5 a, x e k = X k , and A X k = ek for k = 1, 2,
...
Proof. P u t
respectively. Hence IIx’ll 5 E .
80 T
Pal t 1. Theory of Operator Idcals i
h e r e exists up E E' with ( x k , ak) = 1 and Ilakli = ~ ~ k. Define the operator A by that !ail A jakl = o for i
+
~
Moreover, ~ ~ ~ - we 1 may . suppose
W
A := 2'
@
ek
e
1
Using the same technique as above we obtain jlAlj
Ia.
L e m m a 2. Let S E B(E, b'). Suppose that lim 8xk = o wheneoer ( x k ) -is 6
= o crnd lim PmSxk = o for sequence such that lim PTnxk h k wpproxim a 61e.
nb
= 1, 2,
...
c7
bounded
Then S i s
Proof. Assume that X is not approximable. Then we have g := inf jjS
-
PnS,j > 0.
n
Choose x, E E with
[!Sxn- PnSx,/l 2 e/l" and llxnll = 1 for n
= 1, 2,
. ..
By a diagonalization process we select a subsequence (xnh)such that (Pmxn,) and (Pmi3xnk) are Cauchy sequences for m = 1,2, ... Using the method of 1.6.3 we can prove that (Sx,,) is norm convergent. Put y := lim Sx,,.It follows from k
IISxn, - P m S x n , / l 2 that IIy - Pmyll 2 el2 for m
2 @ / 2 for
- Pn,Sx,,jl = 1, 5.
%k
2 wz
. . . This is a contradiction.
We are now prepared t o prove
x
L e m m a 3. Let S E B(E, F ) . If S .is not approxiwiable, then there are q r a t o r s € e ( E , E ) and B E B(F, F ) such that BSXek = ek for k = 1,2, ...
Proof. Without loss of generality we may suppose that ljS\l= 1. According to Lemma 2 there are a sequence (x,) and a number E > 0 such that lim Pmxn= o and
im PmSxn= o for m
= 1,
2, ... as well as llxnll 5 1and /iSxnll2 E
11
Put
&k
n
> 0 for n = 1,2,...
:= &/Zk+l and y, := Sx,. If n, := 1, then there exists m, with
Ikn1
- P m , ~ n , / I5 4 2
and
I i ~ n ,-
pmlynll/ 5 4 2 .
Let n, < -..< nk+ and m 1 < ..-< m k - 1 be chosen. Then we can find nk > n k - , such that
d
~ ~ p i n h - l ~ nEk/2 ~ ~ ~
and
5
~ ~ ~ m 2 - l E k~ / 2n* ~ ~ ~
Now take mk > mk-, with !/Enk - Pmkxn,.!I
2 Fk/2
and
Ikne - pm,!/n,.ll
5 Ek/2*
So we obtain sequences (x,~)and (yn,). Set
x i := x,,
and xi := Pmkxn,- Pmk-lxnh,
y i := Yn,
and
YE :=Pm,Yn,
- Pm,-,yn,
>
5 . Closed Operator Ideals on Classical Banach Spaces
81
where Pm,:= 0. Then arid
-+
I14 - 4II 5 lkrck - Z.'mir,cr~i V'rnB-,x,Ln;i 5 ~k
1 IIY ~ ~2112 I I Y ~-~ ~PrnLyna,~ + iIPrn,-,ynkIIS & k . Moreover, since E 5 li.zi/l 5 1 and E 5 Ily:l] 5 1, we have
3~/4 I I J $/ - /Ix;
-~$1
5 1ldJ15 l l ~ f k l l+ /iQ - ~ i i 2i
1
t
E
and
344 5 11y:i
-
1 0
hy, -
?/:.I1 5 ly:ll 5 lY;ll t I]!/; - y:1, 5 1 t E . A lyil = o for ?L =j= k. Therefore, by Leniina 1 , there are
Clearly A = o and Iy:] operators X,, A, E 2(E,E ) and Yo ,Bo E 2(F , F ) such that
Xoek =- .I-:, A , x ~= e, , and as
Y#?k = g:,
Boy!
== e k
wcll as
I!Xo]l5 1
+
E,
llAoJl5 4 / 3 ~and
/ ] Y o5 /] 1
+
E,
l/Bol'5 4/36.
Since
2
m
11z: - z i 1 1 E~/ 2
and
2 ~ l yi yilj I &/2,
1
1
by setting m
X I :=
61
ek
8 (2;- xi)
and
Y , :=
1
ek
(3(y? - y:j
1
we obtain operators X, E B(E,F ) and Y, t 2(E,P ) with &XI\]5 612 and l]Y,/i5 $2. If X := X , X, and Y := Yo Y,, then S P=~xj and Y p k = y.; Moreover, IlB,Y,\( 5 213 implies the existence of B := ( I $ BOYlj-' B,. Kext it follows from BOY,= I, that BY = I,. Therefore
+
+
BSXq
= BSxk = By: = Bye/, = en.
5.1.2. We now easily obtain the
Ip). Theorem. If 1 5 q < I ,< cn3, then 2(Zp,2), = S(&, Proof. Suppose that there exists S 2(Ip,Z,,) which is not approximable. According to 5.1.1 (Lemma 3) we can find X E 2(Zp,l p ) and B E 2(7,, I,,) with BSXen = e,. Then it follows that
But this is impossible. R e m a r k . We also have e(c,, 2),
5.2.
= S(co$ Zqj for 1
5 Q < oc.
Simple Banach Spaces
52.1. A Banach space I2 is called simple if the algebra B(E, E ) contains one and only one non-trivial closed ideal. 6 Pictach, Operator
82
Part 1. Theory of Operator Ideals
5.2.2. We now give the only examples which are known until now.
T h e o r e in. The Bamch spaces Zp ( 1 5 p
< co)a
d c,, are simple.
Proof. Let E denote either Zp or c,. Suppose that the closed ideal U ( E ,E ) contains a n operator S which is not approximable. By 5.1.1 (Leinma 3) there are operators X , A € B(E, E ) with AXXe, = ek. This means that IE = ASX € %(A!,E ) . Consequently U(E, E ) = P(E, E ) . Therefore B(E, E ) is the only non-trivial closed ideal in 2 ( E , E ) .
5.2.3. Finally, we formulate a n open question. C o n j e c t u r e 1. Every simple Banach space is isomorphic either to l p (1 or to c,.
5 p < 00)
Con j ec t u r e 3. Every simple Banach space is separable.
5.3.
Non-Simple Banach Spaces
5.3.1. We begin with a n elementary result. The Cartesian product El corresponding operators are defined in B.4.11.
x E, and the
P r o p o s i t i o n . Let E := El x E,. Every ideal %(E,E ) tk uniquely d e t e r m i d by the components
U(Ei7 Ek) := {Sik € B(Ei, Ek): JkSjkQi € % ( E ,E ) )
7
where i = 1,2, and k
=
1,2.
We then write
5.3.2. Now a n interesting example is given. T h e o r e m . Let 1 closed ideals :
5 p < q < CQ. Then 2(lpx lq, lp x I,) conta,ins the fdlmuing
Purther non-trivial closed ideals can mdy occur between the ideals connected by i l i ~ broken arrow.
5. Closed Operator Ideals on Classical Banach Spaces
83
Proof. Let %(I, x I,, ,Z x I,) be a non-trivial closed ideal. Then %(Ip, ),Z and I,) are closed ideals in i?(Zp, I,) and B(Z,, Z,) respectively. Moreover, it follows I,) = B(Z,, I,) = @(I,, Z,). from 5.1.2 that %(Z,
%(Z,
We claim that %(I,, 1,) = B(Zp, I,) implies %(Ip, la) = B(Zp, I,). To prove this assertion we denote the identity map of I, by Ipp.Then JpIppQpE %(I, x I,, 1, x I,), where J p and Qp are defined as in B.4.11. Hence, for S,, E B(Zp, I,), we have JqSpqQp
= (JqSpqQp)
(JpIppQp)
c
X
b, zp X
zq)-
I,). Analogously we can see that %(Z,,
This shows that S,, E %(Z, plies %(Ip, I,) = S(Zp,I,).
x I,, Zp x )2, different
We now observe that every non-trivial closed ideal %(Z, from
(
QVp,
w p
I,), > I,),
@G,,
I,, I,)
)
w,>
or
("'".~,), w p >
41,
W q ,
5))
QVq,
I,)
I,) = g(l,, 2,) im-
is of the form
QV,,
(
w p ,
1.
@U,> 1,) 44, @(I,, 4)
ZpL
The two obvious cases are U(Z,, I,) = i?(Zp, I,) and %(Zp, I,) = Q(I,,
Z,).
R e m a r k . We obtain another non-trivial closed ideal %(Zp x I,, Zp x I,) if a(&,Z,) is defined to be the closure of the set of those operators S,, E %(Ip, Z 4 ) which factor through the identity map I,, from 1, into 1,; cf. V. D. MILMAN[l].
5.3.3. The preceding theorem leads to the open P r o b l e m . Let 1 5 p < q < 00. Does the algebra B(Z, nitely many different closed ideals?
x Z, Zp x I,) contain infi-
5.3.4. Let M be a Cartesian Banach space. Then we put
QM :=
rp,where vM
denotes the ideal of M-factorable operators; cf. 2.2.3.
to]form a subbase of open sets. Observe that I'(a) becomes a. compact Hausdorff space. Moreover, C(I'(a)) is R Banach space belonging to Car (C[O, I]). We know froin C. BESSAGA/A. PELCZYrliSKI 113 that C(l'(a)) C ( r ( P ) )implies > ,9, where Q is the first infinite ordinal. Therefore the Banach spaces
(e
N
C ( T ( w ) )C(T(OY)), , . . .) C(l'(0"
",), ...
are mutually non-equivalent. Since there are uncountably many ordinals of the first type, the assertion follows froin 5.3.8.
5. Closed ODerator Ideals on Classical Banach Spaces
85
5.3.12. Finally, we state a n open
P r o b l e m . How many closed ideals arc contained in the algebra 2(Ll[0,1], L1[O,l])?
5.4.
Closed Operator Ideals on Hilbert Spaces
5.4.1. An operator S E e ( E ,P) is called an N,-Hilbert operator if S = Y A with A E 2 ( E , H ) and Y E e ( H , F ) , where H is a Hilbert space such that dim ( H ) < N,
The class of all N,-Hilbert operators is denoted by 8.. R e m a r k . Obviously $jo = 5.I n 6.6.1 we will define $j :=
u &. Y.
5.4.2. By the same technique as used in 2.2.2 we obtain the
Theorem. $, is an operator ideal. 5.4.3. Let H, be the class of all Banach spaces which are isomorphic to some Hilbert space H with dim ( H ) < N,. 5.4.4. Theorem. H, = Space (&).
Proof. If E E Space (&). Then E is isomorphic to a (complemented) subspace M of some Hilbert space H with dim ( H ) < Na.Now i t follows from dim ( M ) 5 dim ( H ) that E E H,. Consequently we have Space (if&) E Ha.The converse inclusion is trivial. 5.4.5. P r o p o s i t i o n . If
LY
< B, then H, c H,. Moreover, sjb c 8,and $:lo' c $;l"s
Proof. By 5.4.4 and 4.2.8 the assumption SjS = $jaor H, = H,, which is obviously false.
&:lo'
= $j;loS would imply
5.4.6. We now prove the main result of this section.
Theorem. On the class of all Hilbert spaces there are only the closed operator ideals and 2.
8,C'OS
P r o o f . Let U be any closed operator ideal. If Space (a) contains all Hilbert spaces, then = 2.Otherwise put
8, := inf (dim ( H ) :H
6 Space (a)).
We now consider an operator So E U(L,,(Q,p), L,(Q, p ) ) such that Sof = sf, where 8 E L,(R, p). Let Q E := ( m E R : is(w)l 2 E ) for e > 0. The restriction of p to Q, is denoted by p,. Furthermore, we define the injection J , E S(L,(Q,,p , ) , L&2, p)) and the surjection &, E 2(L2(Q,p ) , L,(Qc,p E ) )in a canonical way. Then S , := Q,S,JJ, is = S,S;l E a. Consequently dim (Lz(Qe, p E ) )< N,. invertible, and we have IL,(n,,pF) By setting L, := SoqJ,Q,we now obtain a n operator L, E Bnsuch that ll8, - L J 5 F . Hence So E $Elos. Let S E % ( H , K ) . Using the notation of D.3.1 we see that SoE a(L,(Q,p), 2;,(R, p)). s o 8, E ~ ~ ' " s ( L , (ps)Z, L,(Q, , p ) ) and therefore S E $Sj",los((H, K ) . This proves that ?I 2 #?. The convers? inclusion is implied by 8. %.
86
P a r t 1. Theory of Operator Ideals
5.5.
Notes
The theory presented in this chapter has been initiated by the classical paper of J. W. CALKIK [l]. The main result, stating the simplicity of the sequence spaces I , with 1 5 p < 00 and e,, was proved by I. C. GOHBERG/A. S. MARKUS/I.A. FELDMAN [i], See also R. H. HERMAN 111. The given examples of non-simple Banach spares are based on the work of C. BESSA GB/ A. PELCZYBSKI [l], J. LINDENSTRAUSS/A. PELCZY~SKI [l]. H. P. ROSENTHAL [ 2 ] , [3], and G . SCHECHTMAN [l]. The reader is also referred to V. D. MILMAN113. Closed operator ideals o n a fixed Hilbert space of arbitrary dimension were investigated by B. GnAniScH [I] and E. LUFT [l]. Recommendations for flirther reading:
E. LACEY!R. J. WHITLEY[i], H. R. PITT[l], H. PORTA [i], [2], [3], H. P. ROSENTHAL [l], P. VOLIWANN[i].
Quasi-Normed Operator Ideals on Banach Spaces
6.
'The tiieoiy of noriiicd operator ideals has been founded by A. GROTIIENUIEOK and Ic*riltorideal there exists a lot, of cliffcrcnt quasi-norms. However, t hc ,.iiice" quasi-norms are selected by the conipletenebs of the corresponding topology. Prom this point of view for every operator ideal t,hrre is, up to equivalence, at most one rc.aaonablc quasi-norm. Because of this fact it swnis convenient t,o includr the cvmpletcness into the definition of quasi-normed operator ideals.
6.1.
Quasi-Normed Operator Ideals
6.1.1. Let U be an operator ideal. A map A from U into W+ is called a quasi-norm if the following conditions arc satisfied :
(QOI,) A(1,) = 1, where X denotes the 1-diitiensional Banach space. (QOI,) There exists a constant x 2 1 such that B(S1
+ 8,)5 x[A(Sl) + A(S,)]
for A"], 8,E U(E,F ) .
(QOI,) If 7' € f?(EO,E ) , S E g ( E , F ) , and R E Z ( F , Po),then
A(RS7')5 jlBl] A(S)IITII. R e m a r k . Quasi-norms on operator ideals will he denoted by bold capital Roman letters. 6.1.2. Propositioii. Let 8 be an operator irEal with a qumi-norm A . Then all corttpcrnents %(E, F ) are linear l o p o l o c ~ Hausdorff l spaces.
P r o o f . Obvioiisly we have A(U)= II) A($) for S E a(#,F ) and il X . Thus A is a quasi-norm in the sense of LKi)T, p. 1621and we niny generate the corresponding t 0pology. Rcniark. We refer to this topology if we use concel)ts like A-lim, A-Cauchy wquence, A-bounded scit . etc.
6.1.3. A quasi-norwed opercctor ideal ['u, A ] is an operat,or ideal U with a quasi-norm -1such that all linear topological Hausdorff s p w s 9J(E,F ) are complete. 6.1.4. P r o p o s i t i o n . Let [VJ. A] be u quasi-normed operaior ideal. Il'k~nIlSIl 2 A ( S ) for ccll S U.
90
Part 2. Theory of Quasi-Piormed Operator Ideals
Proof. Given S E U(E,3'). Let z t U , and 6 E Up.. Then it follows from = (6 @) 1)S(1@) z) that
(Sx,b) I x
I(fk-915 IF 0 I l l WJ) 1110511 5 -4(X). Hence llSll 5 A(&). 6.1.5. P r o p o s i t i o n . Let [%, A] be a qzcnsi-norttux? o p r o f o r d e a l . T h e n we hncc M a o 0yo) = llaol///yoilfor all a, E E' and yo E F . Proof. We know from B.1.2 and 6.1.4 that llaoil IIYd = /la00Yo11
5 A(a0 @ Yo).
On the other hand. the diagrain a0
E
\Ryo
1
@3 1
a,
+F 1 0 Yo
c
I
s-
+%-
I x
implies A(ao0yo) 5 111 Q yo;:A ( I x )
I/ao@
111 = ~ ~ l]y0\i. u o ~ ~
6.1.6. Next we state a fundamental result.
[a, A] and [d,B] be quasi-nornzed operator ideals. If e 2 0 swch that B(S) 2 eA(S) for all S E U.
T h e o r e m . Let there is a constant
9z
d,then
Proof. Supposethat such a constant does not exist. Thenwe can find S, E U(E,, F,,) such that and B(S,,) 1n for n = 1,3,...
A(&,) 5 (2x)-"
Put E := Z2(En)and F := Z2(Fn)as defined in C.4.1. Since I
k
A f h i l
\
m
.JnSnQn)5 C xiA(Sh,,)2 ( i ? ~ ) - ~ . i=l
k
00
the partial sums 2 JnSnQ,forin a n A-Cauchy sequence. Consequent,lyS := 2' J,S,Q, 1
belongs to 8. Now it follows froin S E 8 that n 2 B(S,) = B(&,SJ,) is a contradiction.
1
5 B(S), which
6.1.7. Let U be an operator ideal. The quasi-norms A, and A2 defined on 41 are called equivalent if there exists a constant e 2 0 such that Al(S)5 eA2(S) and A2(S) 5 eAl(S) for all S E U. 6.1.8. 4 s a n iniinediate consequence of 6.1.6 it follows the iniportant
T h e o r e m . For every quasi-?tornbed operator iden1 [?I, A] the quasi-norm A i s wzique up to equivalence.
6. Quasi-Normed Operator Ideals on Banach Spaces
91
6.1.9. Unfortunately, a quasi-norm A need not be continuous in its own topology. We only know that S = A-lim S , implies n
A(&) 5 x lim inf A(&) n
and lim sup A(Sn) 5 x A ( S ) . n
An easy counterexample is the following. Let
x
> 1 and put
A(S) := !I811 for S E (6 and A(S) := 5t llSll for S
48 .
Then A is a quasi-norm on 2 with
+ S2)5 x[A(S,) + A@',)]
B(8,
for S,, S, E %(E,3').
However, if E is infinite dimensional and S E @(E,E ) , it follows that
-4(S A(S) = llS/l and lim E'O
+ &IE)= x IISll.
This quasi-norm catastrophe is very unpleasant, but not critical. Indeed, we see from 6.2.5 that every quasi-normed operator ideal admits an equivalent quasi-norm which is automatically continuous.
6.2.
p-Normed Operator Ideals
6.2.1. A quasi-norin A on the operator ideal % is said to be a p-norm (0 < p 5 1) if the p-triangle inequality holds :
A(Sl
+
S,)P
5 A(Sl)P
+ A(S,P
for S,, S,E
a(#,F ) .
If p = 1, then A is simply called a norm. R e m a r k . The constant x := 2llp-l can be used to satisfy condition (QOI,). R e m a r k . Obviously every p-norm A is continuous in its own topology.
6.2.2. A p-normed operator ideal [a,A] is a n operator ideal 91 with a p-norm A such that all linear topological Hausdorff spaces % ( E ,F ) are complete. If I, = 1, then [a,A] is said t o be a normed operator ideal. R e m a r k . Every closed operator ideal 91 is a normed operator ideal respect to the operator norm.
[a,11.111 with
6.2.3. We now formulate an important criterion which will be permanently used in the sequel. T h e o r e m . Let % be a subclass of 2 with a n 2*-mlued functim A such that the are .ratisfierJ (0 < p 5 1):
f o ~ ~ o ~ uconditions irq
u
(0) I,y- € and A ( I x ) = 1. co (1) I t Jollowsfrom S,, S,, ... E %(E,F ) and 2 A(S,)P CQ
and A ( S p
5 2' -4(S,)p.
1
< c*3 that8 :=
m
SnE %(E,P) 1
1
T E 2(E,, E ) , S E U ( E ,F ) , and R E 2 ( F , F,) imp&J RST E %(E,, F,) and d ( R S T )S IlBIl A(S) IITII. (3)
T1~e-n[a,A] i-sa p-normed operator ideal.
92
Part 2. Theory of Quaai-Normed Operator ‘Ideals
Proof. The only point is to observe that (1) summarizes the p-triangle inequality and the completeness, as well. 6.2.4. We now consider a quasi-normed operator ideal [a,A], where SL denotes the constant in the quasi-triangle inequality (QOI,). Define the exponent p by 2l’P-l = z. Since SL 2 1 , we have 0 < p 5 1.
Lemma. Let Si E U(E,F ) and ki E (0:1,
...). Suppose
that
A(Si)P
5 2-Od and
II
2-kl = 1. Otherwise,
Proof. Without loss of generality we may sssunie that choose k,+l,
..., k,
m
F,
1
{ O , 1, ...) with 2 2-k6 = 1and put Si := 0 for i = n
+ 1, ...,
?)I.
1
We prove the lemnia by induction. The case inax {kl, ..., k,} = 0 is trivial. S u p p o s ~ ~ that the assertion is true for max (k17..., k,) = h. Xow let niax {k,,..., k,} = h 1.
+
n
2’ 2-kl = 1, the set I = (i:ki = h + 1) contains a n even number of Writing I = { i , ,...,&,) we obtain
Since
indices.
1
for u = 1, ..., 1. Consequently the operator
n
2 Si
can also bc represented in the
1
sn
n,
form
2 Ti such
that -4(Ti)p5 2-kj and
1
2 - k ~= 1 , but max (hl,
...,h,,) = h.
1
Therefore, by hypothesis, it follows that A P r o p o s i t i o n . If Si E %(E,F ) , then
n
1 A(AS,),5 1/2, and choose k , E ( 0 ,1, ...} such that I n n 2-k*5 1. Now it follows froiri 2-kp1 2 A(S;)P 5 2-kf.Since 2’ 2 - k 4 5 1/2, we have P r o o f . Suppose that
1
thp preceding Ienima that A
(: 1
1
S , 5 1. This provcs the assertion.
6.2.5. Finally, we show that every quasi-nonned operator ideal is indeed a p-nornied opcratm ideal.
Theorem. Let vale& p-norm A,.
[a,A] be a
quasi-no,med operalor idenl. Then there exists a n equi-
6. Quasi-Normed Operator ldeala on Iianach Spaces
Proof. Compute the number p from
ic
99
= Z l l P - l , and put
n
Obviously A,(& 5 A(&).Since S = 2’ Si implies 1
we have I/Slj5 Ap(S).Hence A p ( I x ) = 1. The p-triangle inequality follows irnmediately froin thc definition of Ap, and condition (QOI,) is fulfilled, as well. Using 6.2.4 we get A(S) 2 21’pAp(S).This proves. together with A p ( S )5 A(S), the equivalence Qf A and A,.
6.3.
Nuclear Operators
6.3.1. An operator S E e ( E , E’) is called 7~uclenrif m
s = :cg?yi, 0,
1
with a,, a2,... t E‘ and y,, y2, ... E F such that 05
x IbJI ll?/lll
0. Consider a nuclear representation m
S = ,Z ai 0Yi 1
such that 8
,Z llaill llyill I (1 1
+
E)
N(4
-
If T E E(Eo,E ) and R E E(F, Po),then M
RST
= ,Z T ' u ~ @
Zyi
1
and m
2' IIT'aill JPYill 1
I (1
+
E)
IIRIl N(S) IITll-
This proves that RST E %(Eo,Po)and N(RST) 5 (1
+
E)
(jRIlN(S) IlTll.
6.3.3. We now state an import,ant characterization. Theorem. An qerator S E E(E,P) is nuclear i f and only i f there exists a commutative diagram
E
S
6. Quasi-Normed Operator Ideals on Banach Spaces
95
such that So E B(lm, 11) is a diagonal operator of the form So(E,)= (oiti)with (a5)E E l . Furthermore, A E B(E)1,) nnd Y E B(ll, F ) . In this cuse,
N(S) = inf llyll llSollIIAli, where the infimum i s taken over alE possible jactorizations. Proof. Let S E %(E,F ) and E
> 0. Consider a nuclear representation
m
r 9 = 2 u i @ yi 1
such that
21 Ibill IIYtll 5 (1 + E ) “8) 00
ai
Put a! := -,
1~: := L -!
IIYiII ’
llaall
m
*
and ai := llaill IIyill. Next, by A z := ((2,a!)) and
Y(yi) := 2 qiyp we define operators A E B(E,I,)
and Y E f$,,
F ) . Then /[All= 1
m
1
and llYl1 = 1. Obviously IISoil = 2 ai. Consequently, there exists a factorization 1
S
=
YSoA such that jjYII llSoll[IAll 2 (1
+
E)
N(S).
Conversely, every diagonal operator So(ti)= (oJi) with (ai) E I, belongs to ?JZ(Zm, Zl), since M
So = 2 olei @ ei . 1 m
Moreover, ?i(So)5 2 loi/= IISoll. Therefore S = YS& is nuclear and we have W) I IIYII llS0ll IIAIL
6.3.4. I n the proof of the following lemma we consider a n important example of a non-trivial nuclear representation. Here I denotes the identity map. Lemma. N(1: 1;” --f
EL) = 1.
P r o o f . Let dn be the set of all n-tuples e = ( q )with
E~
= -j= 1.
Then
96
Part 2. Theory of Quasi-Normed Operator Ideals
Then
N(S,) 5
SUP 1.rig:n
I c ~ N(I: / 2'; -+ ZL) 5 SUP
10il.
I
Analogously we get
N(S, - 8,)
sup \ci\ for
nL
> n.
n d l m
Therefore (S,) is a n N-Cauchy sequence. Since S is the only possible limit, it follows that S E ?X&, Z-). Moreover,
N(X) = lim N(8,)
4 sup / O i l .
R
1
Conversely, we have sup IUiJ = IJSllg N(8). a
6.5.6. We now consider another important example. P r o p o s i t i o n . An operator S E 2(Zl,Zl) i s nuclear if and cmly if there is an infinite ( o i k ) such that
mcltriz
In thi.9 case, a3
Y(S) = ,r sup 1=1
Id,k/.
k
Proof. Suppose that S admits the described representation. By setting 8
2 o,~&
(2,a,>:=
for r
=
(&)
E I,
k - I
m
we define furictioiials a, E I , with /ju,(ioo= sup joIk{. Since S = k that S E %(Zl, Zl) and m
W S )I s Ib&ICJ lk,ill = 1
CJ
csup I
N
1
such that
bzkl
*
-1 k
Conversely, let S E R(2,, Zl). Given S=xai@yj
a , @ ei,it follows 1
E
> 0,consider a nuclear rcpresentation
6. Quasi-Normed Operator Ideals on Baiach Spaces
6.4.
97
Integral Operators
(i.i.1. An operator S
E B(E, F ) is called integral if there is a constant u 2 0 such that
/trace ( S L ) J5 c IILjl for all L E g ( F , E ) .
We put
I(S) := inf c. The class of all integral operators is denoted by 3. 6.4.2. T h e o r e m .
[a, I] is a normed
operator idecil.
Proof. We use criterion 6.2.3. (0) It follows from I, = 1 @ 1 that , 5 1[Ll1 for L c [trace(Ixh)1 = / ( L ( l )I)/ Hence 1% E 3 and I(1,)
5 1. Obviously 1 5 I(1Ap). b?
(1) Let S,, S,, ... F, 3 ( E ,F ) such that for every finite operator m
L=
g ( X ,3').
bj '3xi
1
we obtain
m
m
2' I(&) < co, and put AS:= 2 8,. 1
1
Then
98
Part 2. Theory of Quasi-Normed Operator Ideals
6.4.3. L e m m a . I(I: 1; -+ 1;) = 1. Proof. Using the representation
I = 2-"X e @ e cv"
described in 6.3.4 we obtain
I ( I : 1;
--f
5 2-"
1:)
I(e @ e ) = 1 . 8"
The converse estimate is trivial. 6.4.4. Finally we consider a n important example. P r o p o s i t i o n . The embedding map I froml, into I , i s integral. Moreover, we have I(I:1, +lm) = 1. Proof. P u t &,(51,
7
.
-,En, En+,,
.-.) := (51,.- *,
6n)
and
Jn(L ..., L):= (51,
...I
En, 0, ...).
Then 6.4.3 implies ltraee (JnQnL)/2 l/L\/ for all L
g(Zm,Z1).
Since trace ( I L )= lim trace (Jn&,L), we obtain 98
]trace ( I L ) /5 I/L\I for all L E &(la,l l ) . This proves that I E 3(Zl,I,)
6.5.
and I(I: I, -+1,)
i1.
Absolutely Summing Operators
6.5.1. An operator S E B(E,F ) is called absolutely summing if there is a constant cs 2 0 such that rn
for every fipite family of elements z,,
..., z, E E .
We put
P(S) := inf
G.
The class of all absolutely suinining operators is denoted by 'p. 6.5.2. Theorem.
[q,PI i s a normed operator ideal.
P r o o f . We use criterion 6.2.3. (0) Obviously 1% E 'p and P ( I x ) = 1.
99
6. Quasi-Xormed Operator Ideals on Banach Spaces
(1) Let X,,S,, ... E @(E, F ) such that xl, . . ., x, E E , then
m
m
2 P(S,)
0, we choose yl,..., y,
E
5 L7F
n
UF
s u (Y*+ SUFI, 1
+
8
where 6 := -. Then [lLll 5 (1 E ) sup IILy,lj for all L E B(F, E ) . By setting I+& a ((Lyi),Ao):= trace (SL) we define a functional A , on the linear subset M := ( ( L y i ) :L E B(F, E ) ]of Z",(E). Moreover,
I((k/i)> A0)I
=
/trace (S-41
s I(&)llLll 5 (1 4-
E)
IfS)SUP lILYz11-
By the Hahn-Banach theorem there exists an extension A such that l((zn), A)I
5 (1
+
E)
I(S) SUP IlzeII for all (J,) E z",(E)-
This functional admits a representation 71
((xJ, A ) = 2 '( x d ,ai) with
..., u,
CI~,
E E',
1
n
and we have
2'
11aJ 5
(1
+
E)
I@). Now, if Lo := b @ x, it follows that
1 n
(83, a> = trace (SL,) = ((Lo?/t),A ) =
z
E E and b F'. This means that S
2' a, @ g?.Consequently
(z,
4 (Yz,b )
1
n
for all .E
=
1
n
No(S) 5 2 ' Ilat'l IIyeII 5 (1 1
+
I(#).
So we have seen that W(S) 5 I(S).
6.5.4. We now prove an easy consequence of 6.8.3. L e m m a 1. Let X E g(Eo,E ) nnd S Z(E,3).Then S"(SS)
s I(S)11x1'.
6. Quasi-Normed Operator Idcnls on Bnnach Spaces
103
Proof. Take the factorization X = JX,, where J is the injection from M ( X ) into E. Then it follows from l&Coll = IlXIj that
s I(S)(IX(1.
NO(SX) I NO(SJ) llXoll = I(SJ)!lXol1
Using the same method we can also check the dual L e m m a 2. Let S E 3 ( E , F ) and B E $(F, F,). Then
NO(BS) 5 IIBil I(S). 6.8.5. Finally, we state a uscful inequality.
P r o p o s i t i o n . Let
.6.9.
S E $(E, E ) . Then [trace (S)[ 5 KO@).
Notes
The concept of a quasi-normed operator ideal on the class of Banach spaces was introduced by the author about 1969. However, the main ideas go back t o the theory of tensor products. Indispensable references are the memoir [GRO] and the “RR6sumir de la theorie mbtrique des produits tensoriels topologiques” by the same author. See also I. AMEMIYA/K. SHIQA[l]. Suclear operators in Banach spaces were independently defined by A. GROTHENDIECK [l] and A. F. RUSTON[l], [ 2 ] . The notions of an integral and absolutely summing operator as well as a Hilbert operator are also due to A. GROTHENDIECK. A few results are taken from A. PELCZYNSKI/W. SZLENK[l] and A. PIETSCH r3], [ 3 ] . Recommendations for further reading:
[LOTI. 3. ROSENBEBGER [l].
Products and Quotients of Quasi-Normed Operator Ideals
7.
The purpose of this chapter is to show that products and quotients of quasi-normed operator ideals may he equipped with a quasi-norm, as well. Important examples will be treated in later chapters.
7.1.
Produets of Quasi-Normed Operator Ideals
i.l.l. Let [a.A] and [d, B] be quasi-nonned operator ideals. For every operator N c ( E ,G) belonging to the product 'u o b we put
A o B(S) := inf A(X) B( Y ) , where the infimum is taken over all factorizations S = 2-1' with A- E %(F. C ) a d Y € B(E, F ) . Clearly, ,4" denotes the n-th power of the quasi-norm A. R e m z r k . The product
7.1.2. Theoreni.
[ao $3,
[ao B, A o B] will frequently be written as [U, A] o [%, HI. A o B] is a quasi-normed operator ideal.
P r o o f . Using the method of 3.1.2 the quasi-norm properties of A o B can easily be checked.
To show the eonipleteness, according to 6.2.5, we may suppose that [a,A] and are p-nornied and 2-normed, respectively. Then, using 6.3.2, we can prove that [U o b, A o B] is an r-normed operator ideal, where l i r := lip l/q. Let S, E o B(E,F ) such that
[a,B]
+
bo
A o B(S,)'
< M.
1
Given E > 0, we can find factorizations X, = X,Yn such that the following conditions are satisfied:
+ c) A o B(S,)]'/p, + A o B(S,)]'/Q.
X , E U(M,, -Fj and A(X,) 5 [(l Y , E B(E,M,) and B( Y r L5 ) [( 1 co
Put S :=
3 X,Q,,
N
E)
W
Y :=2 ' J,Y,, and M := Z2(Wn).Then
B( ITn? 5 (1 + c
m
2 A o B(S,)'
) ~
1
1
irnply X E % ( M .F ) and Y E B(E,M ) . Moreover, we have A(X)P 5 (I +. c)r
m
A o B(AY,)~ 1
7. Products and Quotients of Quasi-Nolined Operator Ideals
105
and m
B ( Y ) Q5 (1
+ E)'ZA o B(S,)'. 1
m
Since S :=2 S, has the factorization S = X Y it follows that S E
o
b(E,P)and
1
m
A o B(S)' 5 (1
+ 2' A o B(S,)'. t)'
1
Sow 6.2.3 yields the assertion.
7.1.3. Kext wr give an example which will be treated in 24.6.2.
[m, li.l]]o [3,I] = [a, N]. 7.1.4. Theorem. Let [a,ll.il] and [b,ii.il] be [?l 0 b, li.l;l = [a,11.113 [B, 11.113. Proposition.
nor?iied operakor ideab. TiLen we have
0
Proof. We consider an operator S t B(E, F ) belonging to o b. Then there exists jmtorization S = XI' with X E % ( M , P ) and Y t B(E, M ) , where M is a suitable Banach space. Obviously, we may suppose that IlY;!5 IISil. Sow
3
llflio := max (llfil, IlXfIl) for f E M defines an equivalent norm l].llo on M . Let M , denote the Banach space obtained from M by this change of the norm. Piirtherniore, let XoE31(Mo,P)and YoEB(E,M,) be the operators given by the same rules as X and Y , respectively. It follows from IlXOfli 5 l l f i l o that liXo/]5 1. Moreover,
/IYozllo= niax (IIYsli, IlSxll) 5 liSI/ ll.~ll for all x 5 E implies llYoll 5 l]Sl/.This proves that IlSIl = inf /jX//IIYl;, where the infimumis taken over all admissible factorizations. R e m a r k . We see froin 7.1.2 and the above theorem that the product of closed operator ideals is closed, as well.
7.1.6. A quasi-norined operator ideal
[a,A] is called idempotent if [a,A] = [W. A']].
7.l.G. From 3.1.7 and 7.1.4 we obtain the P r o p o s i t i o n . Y'he normed operator ideals [6,iI.ll1, [52, il.111, [%k (!re idempotent.
7.1.7. P r o p o s i t i o n . T h e norined operator ideal [Sj,HI is idempotent. 7.1.8. Finally, let us mention a fact which follows from later results. P r o p o s i t i o n . The noi,mcl operator ideals poteizt.
[a, S ] . [3. 21, n7d [?$3. PI are nof i d e w
106
Part, 2. Theory of Quasi-Normed Operator Ideals
7.2.
Quotients of Quasi-Normed Operator Ideals
7.2.1. Let
[a,A]
and [b,B] be quasi-normed operator ideals. 3For every operator
X E e ( E , F ) belonging to the left-hand quotient 9l-l o b me put A-l o B(S) := SUP (B(Y8): Y E %(F, Po), A ( Y ) 5 1). Here Fo ranges over all Banach spaces. On the right-hand quotient a o b-1 the expression A o W(S)is definedanalogously. R e m a r k . The quotients [a o 8-1, A o B-11 and [U-l o b, A-l o B] will frequent,ly be written as [a,A] o [b,B]-1 and [a,A]-l o [b,B], respectively. 7.2.2. Theorem. iderJs.
[%-I
o 8, :Z-1 o B] and
[ao b-l, A o B-l] are quasi-normed operator
Proof. The main point is to establish the existence of A-l o B. Therefore let us suppose that the supremum is not finite for some operator S E U-l o b(E,F ) . Then we can find Y, E a ( F , F,) such that
A(Y,) 5 ( 2 ~ ) -and ~ B(Y,S) 2 n for
n=
1, 2,
...
Put, F,, := I2(FVb). Since
k
m
the partial sums 2' J,Y, form a n A-Cauchy sequence. Consequently Y := 2 J,Y, 1
belongs to tradiction.
a, and we obtain
1
.n
B(Y,S)
= B(&,YX)
5 B(YX), which is a con-
Finally, it is easy to check the ideal properties and the completeness, as well.
7.3.
Notes
See the bibliogra,phical notes of Chapter 3.
8.
Quasi-Normed Operator Ideals with Special Properties
I n the following we investigate procedures on the collection of all quasi-normed operator ideals. I n this case the rule
new: [a,A] + [ a n e ~ Anew] , transforms every quasi-normed operator ideal [a,A] into a quasi-normed operator ideal [ a n e w , Ane~v]. I n comparision with Chapter 4 we have of course to take into account some modifications due to the presence of quasi-norms. Most of them, however, are straightforward. Of considerable importance are the procedures min and maz. In particular, we inention the formulas
and J.L. The concept of ultrastability, introduced by D.DACUNHA-CASTELLE a useful tool to prove the niaxiniality of some special quasinormed operator ideals; cf. Chapter 14 and 19-
KRIVINE in 1971, is
8.1.
Procedures
8.1.1. A rule
new:
[a,A] -+
[!Blew,
Anew]
which defines a new quasi-nornied operator ideal normed operator ideal [a,A] is called a procedure. R e m a r k . The quasi-normed operator ideal as [a,Apw.
[anew,
[anew,
Anew] for every quasi-
Anew] will frequently be written
8.1.2. We now list some special properties: (M) If [a,A] S [a,B], then [U, A]new G [S, B P w (monotony). (I) ([a,A]new)new= [U, A]ne!%.for all [a,A] (idenipotence). A monotone and idempotent procedure is a hull procedure if [U, A] 2 [%, A p w m d a kernel procedure if [a.A] 2 [a,A P w for all quasi-normed operator ideals. Impectively .
5.2.
Symmetric Quasi-Normed Operator Ideals
8.2.1. Let
[a,A] be a quasi-normed operator ideal. Then we set
Adm'(S) := A@') for all X and
[?@ml,
Udml(E,F ) ,
Adual] is called the dual quasi-normed operator ideal.
108
Part 2 . Theory of Quasi-Normed operator Ideals
8.2.2. Obviously we have the
Theorem.
[ U d l ~ a l , A d ~ ~ ils] a
quasi-normed operator d e a l .
8.2.3. Since the property (M) is satisfied, we can formulate the T h e o r e m . T h e rule
[a,A]
dual:
--7
[%dual, Adual]
is CL monotone procedure. 8.2.4. il quasi-noriiied operator ideal [a,A] is called symmetric if [a,A] & [a,A]dual. I n case [a,A] = [a,A]dual the quasi-normed operator ideal is said to be coinpletel?y s?jmmetric.
8.3.5. We now consider some examples. P r o p o s i t i o n . The nortned operator ideals [S I] and [ S j , H] are cornpletel~s~jmt~tetrir. Proof. Sccording t,o 19.1.1 and 19.3.8 the assertion is a special case of 19.1.4.
8.3.6. P r o p o s i t i o n . The normed operator ideal pletely symmetric.
[a, N] i s
syvitiietric, but not con1-
P r o o f . The inclusion [a, N] [a,X]dual is evident. On the other hand, T. FIGIEL/ W. B. JOHNSON [ I] have constructed an operator S E ?&dual which fails to be nuclear.
8.2.7. P r o p o s i t i o n . The normed operator ideal [$,P] i s iiot symmetric. Proof. By 6.5.4, the embedding inap I from 1, into I , is absolutely summing. the other hand I 6 ?@dual(ll, 12).
8.3.
Regular Quasi-NormedOperator Ideals
8.3.1. Let
[a,A] be a quasi-normed operator ideal. Then we set
Areg(S)
and
011
[ W g , drep]
:= A(KFS) for all
X t Areg(E, F ) ,
is called the regular hull of
8.3.2. Theorem.
[%reg, Areg]
[a,A].
is a quasi-normed operator idetrl.
P r o o f . Using the method of 4.5.2 we can easily show that The coinpletencss of U..g(E, F ) is evidcrit, a s u-ell.
AIeg
is a quasi-norni.
8.3.3. Corresponding to 4.5.3 we have the Theorein. il'he :tile reg:
[a.8 1 -3
[%reg, Alrg]
i s n hull procedzw. 8.3.4. A quzsi-norrned operator ideal [U, A] is called regzclar if
[a,A] = [Zr, h ] r e g . .
8. Quasi-Normed Operator Ideals with Special Properties
I09
8.3.5, P r o p o s i t i o n . Let [U, A] be a quasi-nolnzed operator ideal. il'tien 18, A]*ua1 i 4 regutur. P r o o f . Use the method of 4.5.6.
8.3.6. The next result is analogous to 4.5.7. P r o p o s i t i o n . A quasi-normed operator idenl if it is regular and symmetric.
completely sywmzetric i f and only
is
5.3.7. We now consider some examples. P r o p o s i t i o n . T h e normed operator ideals [3,I],
[v,PI, and 18,HI ure 7egular.
P r o o f . The regularity of [3, I] and [$j, HI follows from 8.2.5 and 8.3.6. The assertion is trivial for [P, PI; cf. 8.4.9.
5.3.8. By 8.2.6 and 8.3.6 we have the P r o p o s i t i o n . T h e nornzed operator idenl
8.4.
[a, S] i.s no! regular.
Injective Quasi-Xormed Operat,orIdeals
8.1.1. Let
[a,A] be a quasi-normed operator ideal. Then we set
A I ~ J ( S:= ) A(J,S)
for all S E %~"J(F:,F ) ,
and [!@"J, Ainj] is called the injective hull of
[a,A].
8.9.2. T h e o r e m . [ I U ~ Z ~8J1, n ~ 1is CG quasi-norwed operutor ideal Proof. Use the method of 4.6.2.
8.4.3. Corresponding to 1.6.4 we have the T h e o r e m . T h e rule
inj: [a,A] --f [ ~ I I A ~ JI ~, J ]
is a hull procedure.
5.4.4. To prove the next rcsult we need a L e m m a . Let So E 2(E,F,) and S E 2 ( E ,J'). Suppose that Yo has the metric extension property. If ilSoxll 5 ~/S.I$for all x E E , then tl~ereexists an operator B E 2(F, F,) sue?& that So = BS and l\Bl!5 1. Proof. By setting B o y := yo, where y = Sx and yo = S,a, we obtain a welldefined linear map from M ( S ) into F , with ~ ~ 5B 1., Consequently ~ ~ there is a n extension B E B(F, F,) such that So = BS and IIBIl 5 1. P r o p o s i t i o n . Let [a,A] be a quasi-normed operator ideal. Lei So t e ( E ,F,) and 8 E %(E, F ) . I TfllSoxll 5 &!3xll for all x E E , then SoE Um](E, F,) and ,4ln3(SO)5 A(S).
Proof. By the preceding lemma, we can find an operator B E e ( F , PinJ) such that JF,So= BS and IjBIj 5 1. Hence JF,S, E U(E, FYI) arid A(Jp0S,)5 A@'). This roves the assertion.
110
Part 2. Theory of Quasi-Normed Operat,or Ideals
8.4.5. P r o p o s i t i o n . An operator S E B(E, F ) belorup to Rinj if and only if there exists an absolutely summable sequence of fzcnctionak n,, a2, . .. E E' such that m
lIS2ll 5
i(z, ai>l for all
2
E E.
1
In
this case, m
Ninj(S)
2 ]la&,
= inf
1
where the infirmrn is talcen over all admissible sequences ( a d ) .
Proof. Let S E ! R ~ J ( E F ), . Then, given E
> 0, we can find a nuclear representation
m
a j 18 yp
JFS = 1
such that
X1 llaill IlyPil 5 (1 + M
E)
N'W).
Obviously we may suppose that
lly&ll= 1. Then
00
2 llaill 5 (1 + E ) Nbj(S). 1
Conversely, let 00
llSsl/ 5 2 I(%, ai>l for all z E E . 1
00
By setting A z := ((2,ai)) we define a n operator A E B(E, Z,) with N(S) 5 Z Ila& 1
Moreover, IlSzIl 5 IIAzll for all z E E. Now, by 8.4.4, it follows that S E Binj(E, F ) m
and W j ( S )
5
llaill. 1
8.4.6. As an immediate consequence of the preceding result we obtain the L e m m a . Pu'hj(1:l; + 1;) 5 cK. Proof. Khintchine's inequality means that ~2 '2\(z, e)l - ~for all z E 1;.
1 1 ~ 1 15~ ~
6"
Thereforc
Nmj(1: 2:
--f
Z:)
5c
= ~ - ~lleilm X = cK. 8"
8.4.7. We state without proof the following criterion which can be derived from 17.3.2 and 19.2.7.
8. Quasi-NormedOperator Ideals with Special Properties
111
P r o p o s i t i o n . An operator S E B(E,F ) belongs to 3 i n j i f and only if there exists a finite regular Borel measure p on the weakly compact set UE,such that IjSzil 5
1l(x,
a)I dp(a) f o r aZl
x E E.
u,
I n this case,
Iinj(S)
= inf
p( U E , ) ,
d i e r e the infimum i s taken over all admissible Borel measures.
R e m a r k . We have N i n j ( S ) = Iinj(S) for all S E P 8.4.8. A quasi-normed operator ideal
i ;
cf. 10.3.1.
[a,A] is called injective if [g,A] = [N, A]i*ij.
S.1.9. P r o p o s i t i o n . T h e normed operator ideats ['p, P] and
[a,HI are injective.
Proof. The injectivity of [Q, PI is evident. Then there exists a factoriza.tion J F S= Y A , where A E @(E,H ) ,
Let S 6 $jinj(E,F).
Y E B(H, Finj),and jjYlj IlAlj 5 (1 + E ) H h j ( S ) . P u t Ho := Y-l[M(JF)],and denote
the injection from Ho into H by J . Let Q be the orthogonal surjection from H onto H,. Setting A , := QA and Yo := JF'YJ we obtain operators A , E B(E, H,) and Y o E B(H,, F ) with liAoll 5 l/A/\and jlYoll 5 I)YII.Since S = Y,A,, it follows that S E @(E,F ) and H(S) 5 IIY,ll /IA,lI 5 (1 c) H n j ( S ) . This proves the inclusion [$,H p [B, HI.
+
8.4.10. P r o p o s i t i o n . T h e nopmed operator ideals
[a,N] and [a, I] are not injective.
P r o o f . Consider a diagonal operator S(&) = (an(,). Then S E ! 3 1 2 , 12) implies (a,) E I,. On the other hand, it follows from 8.4.6 that S E ainj(Z2, 1,) whenever (a,) E 1,. Therefore %(&, 12) $: W j ( E , , Z2) and S(12,Z2) =+ 3inj(Z2, 1,).
8.5.
Surjective Quasi-Normed Operator Ideals
8.5.1, Let
[a, A] be a quasi-normed operator ideal. Then we set
Asur(S) := -k(SQE)
and
[Usur, B s u r ]
for all 8 E USUr(E, F),
is called the surjective hull of [U, A].
8.52. Theorem.
[Usur, A s ~ r ]is
a quasi-normed operator ideal.
8.5.3. Corresponding to 4.7.4we have the Theorem. T h e rule SUI':
[a,A]
--f
[rUsur > - Asur
1
is ( I hull procedure. s.5.P. The proof of t h e following result is based on the
L e m m a . Let SoE B(Eo, F ) and S E 2(E,F ) . SuppGse that E, has the metric lifting property. If S,(UEo)& S ( U E )and E > 0, then there exists c m operator A B(E',, E ) such that So = S A and /!All _I 1. E.
+
112
Part 2 . Theory of Quasi-Normed Oprrator ldrals
P r o p o s i t i o n . Let
[a,A] be
(I
quasi-normed operator ideal. Let So A m i n I
is CI kernel procedwe. A.6.4. To investigate the following exanrple we need an elementary Leniuia. Let ( o f )i I,. l'hen, given M
E
> 0, there exit& (ei)
N
P r o o f . We only treat the non-trivial case, where
Choose a natural number m with
and put
m
na
co .such that
113
114
Part 3. Theory of Quasi-Normed Operator Ideals
P u t Soo(fi) := (pi20iti)and R ( t j ):= (piti). Then we have the diagram R 4
ly
+fB
SO
It follows from R'A E 6 ( E , Z,) Noreover,
ZI), and Y E E 6(Zl,F ) that S E %mi"(E,F).
So,E %(Zm, 00
,z eT2 bil 5 (1 + &I2 N(S) 5 R(S). Hence [a,N] E [a,XImin G [z, Ilmin.
IIYBlI N W O O ) llE'4 5 IIYll Mll iniplies P i n ( S )
1
Suppose that X E g ( E , Eo), So E 3 ( E o ,Yo), and B E g(Fo,F ) . Then there are factorizations X = XoQ: and B = J$Bo, where N := N ( X ) and M := M ( B ) .Since llXoll = //XI/and llBoil = IlBlI, by 6.8.3, we have
IV(BS0X) 5 W
O S O ~ O = )
I(BosOx0)5
1 1 4 1W O )
11x11.
(* 1
Xow let X E 6 ( E , ,To), So E 3 ( E o ,Fo),and B E @(Po,F ) . Then we can find operators X n E g(E,Eo)and B, E g ( F o ,F ) such that
X = Il.ll-lim X , and B
= ll.]l-limB,.
16
n
Using (*) we obtain
+
N(BnSoXn- B,80Xm) 2 K ( ( B n - B,) Sox',) K(BmSo(Xn- Xm))
5 I P n - B m I I I(&,) IIXnll
+ IlBmIl I(&) Ilxn
-
Xmll.
Hence (BnSoXn)must be an N-Cauchy sequence. Since BSoX = 1I.li-lim BnSoX, n is the only possible limit, it follows that BXoX E %(E,F ) . Moreover,
This provts that
[3,I]m1nC= [a,N].
8.6.5. A quasi-uormed operator ideal [U, A] is called wzininzal if [U. A] = [U, AJmin. 8.6.6. P r o p o s i t i o n . The nornied operator ideal [%, pu'] is minimal. Proof. The assertion follows from 8.6.4.
8.6.7. P r o p o s i t i o n . T h e normed operator ideals [S, I], [!$ PI, I and , [@, €11 are noi nzinimal. P r o o f . Every nrinimal operator ideal consists of approxirnnble operators only. Therefore the assertion is implied by 6.4.4, 6.5.4, and I , 8.
Y. Quasi-Normecl Operator Ideals with Special Properties
8.7.
11.5
Maximal Quasi-Normed Operator Ideals
[a,A] be a quasi-norined operator ideal. Then [ a m a x , Amax] := [8,l].Jl]-' 0 [U, A] 0 [a, ll.lJ]-l is called the maximal huZZ of [a,A4]. 8.7.1. Lct
8.7.2. By 7.2.2 we have the T h e o r e in.
[%ma-\, Am"']
i s a quasi-normed operatoi ideal.
8.7.3. The next result corresponds to 4.9.4. Theorem. The rule
mux: [a,A] -+ [ a m a x ,
-4ma\]
i s a hull procedure.
[a,
8.7.4. Theorem. Let A] be a p-novmed operator ideal. Then S E B(E,F ) belongs to i f and only i f there exists a constant (T 2 0 such that A(BSX) I G /IRll /jXll for all X E g(Eo,E ) and B E g ( F , P o ) , where Eo and Fo are arbitrary Bnnach spaces. I n this case, Amah(S)
:= inf
G.
Proof. If S E %max(E, F ) , then &n&x(S) := sup AlBSX), where the supremum is taken over all X E Q(Eo,E ) and B E Q ( F , Fo) such that liX'lI 2 1 and IIBil 5 1. Hence
A(BSX)5
Amax(S)
IlBll IjXlj for all
X E g ( E o ,E )
and B E g ( F , P o ) .
Conversely, let S E E(E, F ) satisfy the above condition. If X E @(E,, E ) and B E @ ( F ,F o ) , then there are X,, E 8(Eo,E ) and B, E g ( F , Fo) with X = l/.ll-lh X, n and B = ~ ~ . ~ ~Bra. - l iIt m follows from n
A(BnSXn- B,XX,)p
5 A((Bn- B,)
S9,)P
- @[llBn- BdiplI&llp
1A(B,S(X,z -
+ lIB,l?
I\&
X,))'
- Xd'l
that (B,SXn) is an A-Cauchy sequence. Since BSX = ll.~l-hn B,XX, is thc only 1z possible liniit, ~ 7 have e BSX E 'U(Eo,F o ) .Moreover,
A(BSX) = lim A(B,#X;,) =( (r lim llBnIl llXnll = G IIBI!IIXII. IL
Hence S E
'Umax(E,
F ) aiid
11
Ama\(S)
5 (T.
R e m a r k . The following theorem shows that in the above criterion the Banach spaces Eo and Po may supposed to he finite dimensional.
8.7.5. Thvnrciti. Let [a,-41 be a p-normed operator ideal. Then S 5 2 ( E , F ) belongs to a m a ' ( E , E') if t r n d only if tkere exists a constant 5 2 0 such that A(Q;&'Jff) 5 5 for nlZ 31
Dim ( E ) and
N E Cod (3').
116
Pa16 2. Theory of Quasi-No1ined Operator Idcals
I n this case. .imax(Aqj
= inf
(T.
P r o o f . Let S c % - x ( E , finite.
Z'). Then S(QTSJTf)5
Am~~~(S since ),
QZ- and J g arc
Converdely, let X E g(E,, E') and R < g(F,Po). Then theie are factorizations = JCX, and B = BoQT, such that //Xoll= IIXli and ]IB,ji = IIBII,where M := M ( S j and N := A7(B).If the ahore condition is satisfied, then
S
P)and Ama'(S) 5 6,by 8.7.4.
Therefore S f %max(E,
Remark. Obviously [Umax, Amax] is unicply dt+winined if t h e p-norm A is only linown on the collection of all elementary operators.
[a, plT]max = [3. I].
8.7.6. P r o p o s i t i o n .
Proof. Let S E !.l@ax(E, F ) a n d L E g ( F , 3;). T h ~ ntherc is a factorization L = J:&,&$ snch that IILoi\I= \ID/,where 111 := N ( L j and N := N ( L ) . R y 6.8.3 and 6.8.5 we have Itrace ( S L ) /= /trace ( & ~ . S J ~ L ,5) /K o ( QiS J sLo ) =
Hence S
s(Q:,sJE,r;,) 5 N(Q;sJ;I)
1IL,II 5 P
a y s )
ilL\I.
E 3 ( E , F ) and I(S) 5 Nmax(S).
Conversely, let S
N(&:+!!Jif)
3 ( E , P),41 E Diin ( E ) ,ancl N =
c
Cod ( F ) .According to 6.8.3
I(&:.SJ$f)5 J(S).
Vsing 8.7.5 we obtain S E %max(E, F ) and 8.i.7. A qmsi noriried operator ideal
Pax(&')
5 I@).
[a,A] is called nzcrxinzal if
[U, A]
j21,
:
.i]m~11.
8.7.8. We now consider soine examples.
Pi o p o s i t i o n . The norwed operntor ideals [3.I], ib,PI. nnd
Pi oof. The maxiiuality of [3,11 follows from [3, I] Let S E ?@max(E,F ) and x l , ....a , E E. S o w \Srt, b,) = I/Sx,/iand llbLl; 1. By setting
UP
-=
[a,Xjma'.
choose b,,
n
X ( t , ) :-=
ttzL and By := ((y, b , ) ) 1
we define operators XI E
g(Z2,E ) and R f g(F,.);Z
[B, HI m e nzaxrmal.
Jlorcovcr,
..., 6,
t
F'
P W ~t h ~
t
8. Qumi-Normed Operator Ideals with Spccial Properties
117
Xow i t follows that n
7I
Hence S E p ( E , 3') and P(S) 5 P n ~ a x ( S ) .
The inaximality of [SJ, HI is a consequence of 19.3.8.
[a, S ] i s not ?nuximul. Proof. The assertion follows from [a, K] =+ [S, I] = [a, 'N]mm.
4.5.9. P r o p o s i t i o n . T h e norined operator ideal
S.i.10. P r o p o s i t i o n . Let
[a,A]
be a quasi-normed operator ideal. Then
[U, _4]reg & [U,A ] m x . Proof. Use the method of 4.9.10. b.7.11. As an immediate consequence we have the
P r o p o s i t i o n . Every iwzximnl quasi-norrned operator ideal is regular.
8.7.12. P r o p o s i t i o n . Let
([a,A]dual)max
=
[a,A] be a p-normed
operator ideal. T h e n
([U,A p = ) d u a l .
P r o o f . Let S E 2(E,F ) belong to (Umax)dml. If X E g(Eo, E ) and B E g ( F , PO). then A(X'S'B') 5 Amx(S') l(X']]IlB'11.
It follows that AduaI(BSX)
Hence S E
5 ( A m - ) d m l ( S ) lIBl]IlXll.
(Udml)max(E,
P)and (Adual )max ( 8 )5 (Am=)dual(S).
Conversely, let S E 2 ( E , F ) belong to ( U d ~ 1 ) m a x . Suppose that Y s ( P &F ' ) and 14 E 3(E', EL), where the Banach spaces E, and F , are finite din~ensional. Then there is B E $ ( F , Po) such that Y = B'. Obviously llBll = llY11. Given E > 0, by E.3.2, we can find X E $(Eo, E) with AS'B' = X'S'B' and llXll 5 (1 $- E ) IIAll. Since AS'Y = (BSX)',we have
A(AS'Y) = A d l * I ( B S X ) 2
(Bdal)ma(S)
5 (I + F ) S o w 8.7.4 (remark) yields S E
(Umax)dml
(A-1
liB/l ilXll
1 . (8)iiAiI lil'il. Inax
and (-4max)dml(S) 5
(Adml)max(S).
R e m a r k . It is very likely that the above formula also holds for arbitrary quasiiioriued operator ideals.
*
8.7.13. P r o p o s i t i o n . Let [U, A] be a quasi-normed operator ideal. T h e n
([U,A]inj)-
=
([a,A]max)inj.
118
Part 2 . Theory of Quasi-Normed Operator Ideds
Proof. Let S E e ( E , F ) belong to (Urnax)lnj.Suppose that X E B(E,, E ) and R E B(F, F,). Then there csists B, E B(F1*j,F F ) with JFoB= BoJFand JIB,]]5 (1 c) IlBl].So we have the diagram
+
s
P-E
JF
Since JFSE Iumax(E, F i n j ) , iC, follows that
A(JFeBXX)
=
B(B,J,XX)
5 ;lmax(JpS) $?lOii jjSlj.
We obtain
A i n j ( B S X ) 5 (1
+
E)
(_i"ax)inj(S) ]lBll IiXil.
Hence S E ((Uinj)max and (Ainj)max(S) 5 (Amax)inj(S). Conversely, 1t.t S E 2(E,F ) belong to (Uinj)max. Suppose that X E @(E,, E ) and B, E Q(Pinj, F,). By 18.3.2 and 19.3.9 there exists a factorization B,JF = I'B, where B E B(F,, ) ,Z Y E @(Zm, F,), and jJYlj llB!l 5 (1 + E ) [iBoll.So we have the diagram
_
_
BSX
_
_
_
_
_
f
F-,I
Y
Since I , possesses the metric extension property, it follows that
A(B0JFSX)
=
A ( Y B S X ) 5 / / Y /A/ ~ ~ J ( B (= ~ X( - )4 i n ~ ) m x(8) ljYl/ ijBlj j\Xll.
Hence 8 E (Umax)ln~and (Amax)ln~(8)(= (A1nJ)max (8)*
8.7.14. I n a similar way we obtain the
[a,A] be cc quasi-normed opemtor idecrl. The??
P r o p o s i t i o n . Let
([a,A ] s m ) m x
=
([U. A]m-)s~r
.
5.7.15. Using the method of 4.9.12 we may establish the P r o p o s i t i o n . Let [a, A] he n quasi-normed operutor ideal. Y ' h e ) ~
([a, A]mln)m*x
=
[U, A]max
and
([U,A]-)mln
=
[U, Almin.
8. Quasi-Normed Operator Ideals with Special Properties
119
8.7.16. Let A be a quasi-norm on the operator ideal 3. Then A is said to be wenkly lower semi-continuoue if _4(S)5 lim inf A(8,) Y
for every directed family of operators X, E %(E, F ) converging to S E % ( E ,F ) in the weal; operator topology.
8.7.17. As shown in 6.1.9 an arbitrary quasi-norm A need not be lower semi-contii w o u s in its own topology. However, we have the P r o p o s i t i o n . Let [a,A ] be n maximal p-narmed operator ideal. Thez A i s weakly loiuer semi-continuous. Proof. Let X c g(E,, E ) and 23 t 3(F,P o ) ,where E , a i d Fo are finite dimensional. Clearly B(E,, F,) is also finite dimensional. Suppose that the directed family of opertitors X, E % ( E , F ) tends to S E %(E,F ) in the weak operator topology. Then R S X = A-liin BS,X. Consequently
A(BSX) = lim A(BX,X) 5 lim inf A(S,) liBll IlXii. Y
Y
The assertion now follows from 8.7.4 (remark).
5.7.18. It can be deduced from Enflo’s counterexanlple that the nuclear norm Pi is not weakly lower semi-continuous. A much simpler example is given in the following P r o p o s i t i o n . T h e equation
A(S) := llSjl
+ inf (118-
L E z ( E , F ) ) for S E B(E,F )
dffines a norm A on 2 which fails to be weakly lower semi-continuous,
8.8.
Ultrastable Quasi-Normed Operator Ideals
8.8.1. I n the following let (E,) be a fanlily of Banach spaces. Furthermore suppose that a n ultrafilter u is given on the index set I. The Banach space of all bovnded families (xi),where x, E Ei for i E I . is denoted by l w ( E t ,I ) . Moreover, put cu(E,. I ) := (x,)E l w ( E l ,I ) :lim llxLii = 0). U
We now form the quotient space ( E t ) u:= l w ( E t ,I)/c,(E,, I ) . If .XI = ( z z ) U denotes the equivalence class corresponding to (xz),then the norm of z can be computed by
The Banach space ( E z ) uobtained in this way is called tlic ultraproduct of the Banach spaces E , with respect t o the ultrafilte lt.
120
Part 2. Theory of Quasi-Normcd Operator Ideals
8.8.2. Let (a,)E Zm(Ei,I ) . By setting
(x,a) := lim (xz,C I , ) for x = (xi)u U we define a functional a == ( u ~ )on , ~(Ei)uwith \]ail= lim Ilajll. U
R e m a r k . This means that the ultraproduct (E:), is metrically isomorphic to subspace of [(EJ,,]’.
R
8.S.3. Let (E,) and ( F , ) be families of Banach spaces. Suppose that (8,)is a bounded faniily of operators X,c B(Ei,P i ) .By setting
(&)u
(5th
:== ( & 4 l l
we define a n operator (St)llfroin (Bz)tlinto (F,)uwhich is called the ultraproduct of the operators S , with respect to the ultrafilter 11. Moreover, Il(Sj)ull = 1iF liS,ll.
8.8.4. Suppose that S E O(E, F ) . Let I be the set of all indices i = ( M , N ) with Jf E Dim ( E ) and N E Cod ( F ) . Choose a n ultrafilter U containing all subsets {iE I:M 2 M,,N 5 X 0 ] , where io= (&lo, N o ) is fixed. Finally, put E , := N . P,:= FIN, and Si:= QgSJg. We now show how the operator S can be reconstructed from its elementary parts. Leninia. There are operators J E B(E,(E&) and Q € B((Ft)lt,J”’) such that 1, IIQII 2 1, and K 4 = Q(X,), J .
iIJd 2
Proof. The operator J is defined by
Jx Obviously
x if x E M , o if x 4 X.
:= (xi)ll with x i :=
jiJll 2
1. Moreover, let
whwe (yi)E Zw(Pi,I) such that
Q ( ( Y ~ ) := ~ ) F’-lim KFyi, U
Q:yj = yp.
First we check that the right-hand expression does not depend on the special choice = o and y” := F’-lim KFyi. Then of ( y i ) . For this purpose assume that I1
i(KFyi,b)j 5 IiQZ-Yil] JIbilwhenever N is contained in the null space of b E P’.Therefore I(Y”, b)I = 1$ (KFyi, b) 2 l i p IIGyiIi IIbII = 0 -
I
I
Hence y” = 0. This proves that Q is well-defined. If the faniily (yj) is chosen such that sup llyiil 5 (1 E ) II(yp)uil, then we get
+
I
IIQ((yP)u)iI5
Therefore
IIyiII S (1
+
E)
II(yP)uII-
5 1. Finally, observe that
&(LY,)~ Jrc = &(Sirci)u
=
P’-lim KFSJgxi = K,Sx U
for all x E E
I
S. Quasi-Normed operator Ideals with Special Properties
8.8.5. -4 quasi-norined operator ideal (&)I1
(ml)and
E )U((Ei),l,
121
[a,A] is called ultrastable if A((S,)u) 5 linl 11 -4(Si)
for every A-bounded fainily of operators Si E a ( E i ,Fi) and cvery ultrafilter U. R e m a r k . Let [a,A] be a quasi-normed operator ideal. Then we can define t1.c. ultrastable hull [Ndt, A ~ l t ]as the intersection of all ultrastable quasi-normed operatror ideals containing [a.A]. Until now no direct characterization of this hull is known. 8.8.6. We now prow the main result of this section.
[a,A] be nn ultrastable quctsi-norvied [a,A]max = [a,A]reg.
T hpor em. Let
operator ideal. T h e n
Proof. Suppose that S E a m a x ( E , F ) . As in 8.8.4 we put S, := Q;SJ$. Then A(#,) 6 A ~ Y ( QSince ).
it follows that K,S
A(K, S )
= Q(S,)u J
Ei:= M , F , := H / X , and
t % ( E ,F')and
A(Q(S,), J ) 5 -i((St)u) 5 l i p A@,) 5
-@ax(S).
Hence S E % r e g ( E , F) and A l e g ( S ) 5 Amax(#). So [a, A]max 2 inclusion has been checked in 8.7.10.
[a,A p g . The converse
5.8.7. As an immediate consequence we have the
P r o p o s i t ion. Every ultrastable reqular quasi-nornied operator ideal is muximl. R e m a r k . Let us mention that, conversely, every niaxinial p-normed operator ideal is ultrastable and regular; K. D. KURSTEN[ 11.
8.8.8. P r o p o s i t i o n . For every ultrastable quasi-norm& operator ideal the injective hull i s ultrastable, as well. P r o o f . Let [a,A] be a n ultrastable quasi-normed operator ideal. Supposs t h n t (S,) is an -41n~-boundedfamily of operators 8, E UlnJ(Et,F , ) . Since
l l ( ~ t 4 u=l ~'/(JF,Stz,)ui/
for all ( 4 u E
(4h
by 8.4.4, i t follows tjhat (S2)aE 211n~((Ez)U, (F&) and
5 lini A4(JFIS,) = lim A ~ I * J ( S ~ ) . A~~J((5 S ~A((JF,Sg)U) )~) U
11
5.8.9. Csing 8.5.4 the next result can he checked in a similar u7ay. P r o p o s i t ion. For eiisry ultrastable qumi-normed operator ideal the surjective hd1 well.
i s dtrastable, a s
8.8.10. As recently proved by S. HEINRICH [3] we also have the P r o p o s i t i o n . For every ultrastable quasi-nwmed operator ideal the d m 1 quasi?Lorma?oyeratoi ideal i s wltrastable, as wel:.
122
P a r t 2 . Theory of Quasi-Normed Operator Idealr
8.8.11. Finally, we give a n example. P r o p o s i t i o n . The normed operator ideal [@, PI is ultrastable. Proof. Let (8,)be a P-bounded faiiiily of operators S , E @(E,,F t ) . Then we put ( P ( 8 , ):i E I].Now, given ( x ~ .~..,) ( ~x ,, , ~E )(~E , ) u ,we have
a .= sup
for all i
I . Hence
According to 8.8.2, the functional corresponding to (a& belongs to the closed unit is absolutely summing and ball of [(Ei)u]’.Consequently the ultraproduct (8i)n P((Si)u)5 sup {P(S,):i E r).
8.9.
Notes
As stated before the idens of injectivity and surjectivity first appeared in I. STEPHANI[Z], [3]. The corresponding notions in the context of tensor products go back t o A. GROTHENDIECK [4].The most important example, namely the injective hnll of the normed ideal of nuclear operators, was investigated b y A. PIETSCH [6]. Symmetrie, regular, minimal, and maximal quasi-normed operator ideals were studied by A. PIETSCH [13] and H.-U. SCHWARZ [l], [4]. I n a famous paper D. DICUKIIA-CASTELLE/J. L. KRIVIKE [l] observed t h a t the concept of ultrastability is an important tool t o deal with quasi-normed operator ideals. See also [SEMI,. exp. Csj. Recommendations for further reading:
[LOT]. [SEMI,. exp. 7, 81.
S. HEINRICH [3], J. R. HOLUB[2], J. L. KRIVINE[l], I 0 there exists an operator L E g ( E , E ) such that lILl1 5 1 and
/jx - L.qj 5 E whenever
zE
K.
131
10. Bnnach Spaces with Approximation Property
10.2.2. Leillnla. A Banach space E has the metric approximation property i f , given x1, ..., x, E E and E > 0 , there exists a n operator L E g ( E , E ) such that IjLil 5 1 and jlxc - Lr,jJ
for
E
i = 1, ..., n.
Proof. Let K be a compact subset of E. Then we can find xl, ...>5, E E with n
K
U {xi + E UE}. 1
By hypothesis, there is L E g ( E , E ) such that llLil 5 1 and llxi - Lxill 5 i = 1, . . ., ‘12. Let x E K and choose x, with jlx - xkll 5 E . Then jln - Lxlj 5
1 1 5
- xkil
+ i]xk - Lxkll f IjLx,
E
for
- Lx/l 5 3 ~ .
10.2.3. The following result will be proved in 19.2.4 and 19.3.5. T h e o r e m 1. Let (Q, p ) be any measure space and let 1 has the metric approximation property.
p 5
00.
Then L,(Q, p )
T h e o r e m 2. Let K be any compact Hausdorff space. Then C ( K ) has tlzc metric approximation property.
10.2.4. L e m m a . Suppose thatE has the metricapproximationproperty. Let M Dim ( E ) and let E > 0. Then there exists an operator A S ( E ,E ) such that IlAll 5 1 E and Ax=xfmallxEM. 6 Proof. P u t n := din1 ( M ) and choose 6 such that 0 < 6 < 1and n -I E . 1-6Then we can find L E g ( E , E ) with IjLlj 5 1 and 11% - Lxll (= 6 whenever x E U M . Consequently
+
llLxll 2 IlxIl
-
ijx - Lx1/ 2 (1 - 6) ljzlj for all x E M
This means that the restriction of L on M is one-to-one. Therefore dim ( L ( M ) )= n. By B.4.8, there are x l , ..., r, E Jl and a,, ..., a, E E’ such that IiLxill = 1, lla,Jl = 1, and @xi, ak) = cik. P u t
P := 1TE
+
n ak
‘3(xk - Lx,).
1
Then IIn,
-
Lql/ _I 6 ilxkli and (1 - 6) IIxkii 5 llLxkll = 1 iniyly
Since xl, .._. x, is a basis of A1 acd T’Lx, = xl,we have 1’Lx = x for all x E M . Consequently A := PL satisfies the desired conditions. 10.2.5. We now obtain the fundcniental Lemma. Suppose that 3’ has the metric approximation propert!j. Let S $ B(E, F ) a n d let E > 0. Then there exists a n operator B € s ( P , E’) such t h t IIRII S 1 E and BS = S.
+
132
Part 2. Theory of Quasi-Normed Operator Idcalb
10.2.6. Finally we state a dual version of the preceding result. Lcinrna. Suppose that E' has the metric approximation property. Let S 6 g ( E , F ) and let t > 0. Then there exists a n operator X E g ( E , E ) such that llXlj 5 1 E and
+
xx = s.
P r o o f . Choose 6 > 0 with (1 + 6)2 5 1 + E . Applying 10.2.5 to S' we can find A E g ( E ' , E') such that ~~A~~ 5 1 + 6 and AS' = S'. By E.3.2 there exists X E g ( E ,E ) with lIX1i (1 i6) IlAli and X'S' = S'. Consequently I/X//5 (1 4 21 E and SX = A'.
+
Consequences of the Metric Approximation Property
10.3.
10.3.1. We begin with an improvement of 6.8.3. P r o p o s i t i o n . Let E' or F possess the metric approximation property. Then Y(S) = N(8) = I(S)for aZE S 6 g ( E , F ) . Proof. Suppose that E' has the metric approximation propcrty. Given E > 0, by 10.2.6, there exists X E g ( E , E ) such that [\Xi/5 1 E and SX = S. Using 6.8.4 (Lemma 1)we obtain
+
W ( S ) = NO(SX) 5 I(S)I1Lq
s (1 +
E)
I(S1.
Therefore Fio(S)5 I@). This proves that No(S) = S ( S ) = I@').
Tile other case can be treated analogously.
10.3.2. T h e o r e m . Let E possess the metric approximation property. Then every operntor S E %(E,E ) has a well-defined trace which can be computed by Kl
trace (8)= 2 (ai,x i ) , 1 CQ
where S =
aj @
xi i s any nuclear representation. Moreover,
1
jtrace (AS) 1 5 N(S)
.
Proof. By 6.8.5 and 10.3.1 we have Itrace (S)] =( &(AS) for all S E g ( E , E ) . Therefore the functional, S -+trace (S),admits a unique extension on %(B,E ) . Furthermore, a7
if
S = 2 ai@ x i is any nuclear
representation, then
1
/ n
\
a
;
trace (27)= lim trace n
R e m a r k . Obviously the same statement is true if E' possesses the metric approximation property. Indeed, this is a stronger assumption, since E has the metric approximation property if E' does; cf. [DIU, p. 2441 and [GRO, chap. I, p. 1801.
10.3.3. Theoreni. Let [a,A] be (I quasi-normed operator ideal. Suppose that E' and P have the metric approximation property. Then (Umln(E, F ) consists of those operators S E U(E,F ) such that there exists a sequence (8,) of operators S, € g ( E , 3') with S = A-lim S,,. 91
10. Banach Spaces with Approximation Property
133
Proof. Let S f BUmin(E, F ) . Obviously A(S) 5 Amxn(S). We now consider a factorization X = YSoA, where A E Q(E,Eo), So E N(Eo,Fo), and Y E 6 ( F o ,F ) . Then we can find A , < $ ( E , E,) and Y , E g ( F o ,F ) such that
A P u t S,
=
1l.lI-lini A,
and
n
:= YnXoAn.It
Y = l~.ll-limY , . n
follows from S = Amln-lim S, that X = A-lim S,. This n
n
shows, without any assumption on E and F , the necessity of the above condition. We next consider an operator S $(E, F). By 10.2.5 and 10.2.6 there are X E $(E, E ) and B E g ( F , F) such that IlXIl 5 1 + C, IlBIl 5 1 F , and BSX = S. Consequently _4m'n(S) 5 (1 + F ) B(S). ~ This proves that A(&)= Amln(S) for all S g ( E ,F ) .
+
Finally, suppose that S
==
A-lim S,, where Sn E g ( E , F). It follows from n
A(& - 8,) = Amln(S,, - S,,,) that (8,) converges in possible limit, we have S E %mm(E, F ) .
amin.
Since S is the only
R e m a r k . The above statement means that %min(E, F ) is the A-closure of $ ( E , F ) in %(E,3').
10.3.4. T h e o r e m . Let [%, A] be a quasi-normed operator ideal. Suppose that B' and P have the metric: upproximation property. Then ama"(E,F ) consists of those operators S E C(E,F ) such that there exists a n A-bounded directed family (S,) of operators R, E $(E, F ) converging to S it& the weak operator topology. Proof. Let S satisfy the above condition. If M E Dim ( E ) and N E Cod ( F ) ,then = 8-lim Q$S,J$. This implies, by 6.1.9, that
2 ( X ,F I N ) is finite dimensional. Hence we have QZ#JZf A(Q$SJ$) 5 x lim A(Q!$S',J%). Y
Therefore S E U m a x ( E , F ) and Amax(S) 2 x sup A@,). Y
Conversely, let S E F). Since F possesses the iiietric approximation property, we may choose a directed family of operators B, E g(F.8')such that 'lBvl\2 1 and lim Bvy = y fcr all y E F . Given E > 0, by 10.2.6, there are X, E g ( E ,E ) Y
with //Xvll5 1
+ E and B,,SX, = SJ. P u t 8, := R,SX,.
Sx = lirn B,Sx
If x E E , we have
= Iim B,XX,x.
Y
Therefore the directed family (8,)tends t o X in the weak operator topology. Noreover, -I(s,) 5 ilBui'Amax(S)]/X,li5 (1 + E ) *4max(8).
10.3.5. Theorem. Let [%. A] be a n2isirmi normed operator idccil. Xuppso fhrrt E' rtnd P have the metric approximation property. If T t U * ( F ,K " ) , then
(8,T) := trace (A'T) defines a functionctl T on %(E, F )
134
Part 2. Theory of Qmii-Soimed Operator Ideds
The correspondence obtained in this way is a metric isomorphism between [%*(F, E"), A*] and [%(E,F),A]'. P r o o f . Suppose that T < %*(F, E"). First 1r.t AS' g ( E , F ) . Uiveii P > 0. hg 10.2.4 arid 10.2.5, we can find X E g ( E , F:) and B f g ( F . F ) siioh th'it ]]Xi:5 1 f 8. lIBll : 1 F, and B S S S. S o v
+-
1
itrace (XX"TL)i 5 A ( S ) //X"jj A*(/') llLl] for all L E g ( F , F ) , yields I(SXV) 2 A(&) [ / X A*(T).Using 6.8.4 it follows that
+
R'(S"T)= 3(RA'S7T) 5 jjBl/ A(S)IIXII Ai*(!Z1) 5 (1
F)')
A(S)A*(T).
Hence N(S"T) 5 A ( 8 ) A*(T). Since [%, A] I\ miqinml, for every S 6 %(E,F ) there are S,,E g ( E , F ) with S = A-lim S,. Then S(S;JZ- SLT) 5 A@,, - 8,) A*(T) n
implies that)'!$'SA( s o we get SnT
is an 3-Cxuchy sequciici~.i h i t S 1' k the only imssible liniit. R(F,F ) . By 10.3.2. thr traw of S T is well-tl~fiiietl.Moreover.
itrace (S*T)i5 ?i(-(sz/')
5 A(S) A * ( T ) .
Therefore A'(T) 5 A*@), where A'(?') denotes tt-w norm of the corresponding functional. Let now T he a functional on U ( E ,F ) . Then we citn find an operator T E g(17, E") such that ( a @ y, T ) = (Ty, a ) for all a E E' and y E F . Hence ( S , T ) = trace (ST) whenever S E g ( E , F ) . Next we prove that T E %*(F, E"). For this purpose 1Pt Lo E U(E;, To), and Y 5 g(Fo,If'), where Eo and F , are finite A E 8 ( E f fEg), , dimensional. Given F > 0 , by E.3.3, we can find X 5 g(K:. E ' ) such that AT Y = X'T I.' and IlXIl 5 ( 1 F ) IIAlj. Moreover, there exists A, :g ( K , E,) with X = Ah. Then we have A T Y = A:TY and lIA,jl 5 (I t) IiAjI. Henccb
+
+
trace (ATYL,) = trace (AiZIJ-Lo)= trace ( 1 7 L o K ~ o A ~= ? ' (YL&peA,. ) 'p) and Itrace (TYL,A)/ 5 A(PL,KFaA,)A'(T) 5 (1 -k
t)
A'(Y)
This proves that T E %*(E', El') and A*(Z1) 5 A'(T). Siricv finally obtain (AS', T ) = trace (S'T) for all S E %(E. F ) .
[a,A]
is niinimal, n e
10.3.6. We now show that the adjoint norrued operator ideal can be defined in a much simpler way if the underlying Banach spaces have the metric approxiination property. Theorein. Let [a, A] Ire rl n o r i d operator ideal. Szippow t h t E rl?zcl F possess the metric approximation property. Then S E g ( E , F ) beloup to %* if nnd 01?71/zf there exists (1 constavzt (T 2 0 such that
aA(L) for all 1; 6 g(F,E ) .
/trace (SL)i
I n W&& case, A*(& = inf
0.
10. Bsnach Spaces with Approximation Property
135
P r o o f . Let S satisfy thc above condition. Given B E g ( F , P o ) .Lo E U(F,, B0),and X E S(Eo,E),we piit I; := XLoB. Then Itrace (SXLoB)[= Itrace ( S L ) /5 oA(L) 5
(T
jIXl1 4(L0) 1;BjI.
Hence S E 'U*(E,F ) and A*(S) 5 (T. Conversely, let X E %*(I#, 3'). If L E g ( F , E) and E > 0, w-t can successively find X E g ( E , E ) and B E g ( E , E ) such that jlXll 5 1 E , jjBI1 5 1 E , X L = L, and BSXL = SXL. Then SL : XXL = BSXL. Therefore
+
+
+
E ) ~
-
0) Ok
ltrace (SL)I = jtrace (SXLB)j 5 A*(&')llX11 A(L) IiBI: 5 (1
A*@) A ( L ) .
This shows that /trace (SL)j 5 A*@) A ( L )for all I; E g ( F , 12).
10.4.
Counterexamples
10.4.1. We start this section with an elementary Then Lemma. Let 8 E [0,1] and ( E 9.
+ (1 - e) exp (-eg) 5 exp Proof. It is enough to treat the case < 1 0. Using O(1 - O)k f ( I e exp ((1- 6) ()
((2).
5 I . from
and
we obtain {2
(-3
1"4
0exp((1-0)r)+(l-e)exp(-e85)~1+++---+++.. 2! 3! 4. - exp (5) - C 5 exp ( { 2 ) .
10.4.2. Let bmbe the set of all vectors e = ( E ~ ., .., E , ~ )with cI = $ 1 . Pix a number 0 E [O, 11. Then we define a probability ,urnon d m by setting p n * ( ( e ):= ) ea(l - O ) @ for e E brn,where a := card fj:ci = +1] and B := card ( j : ci = - I } . Xoreovcxr, let f j ( e ) :=
1 - 0 for - 0 for
ci = $1, ci =
-1.
We now prove a generalization of E.5.2. Lemma. Let z = (Cl,
...,),[
E 1; and 0 < p < 00. Then
where co i s some positive constant.
136
Part 2. Theory of Qiinsi-Normed Operator Ideals
P r o o f . Observe that the randoin variables f l , case, by 10.4.1, WB have
..., f m
Sow choose a nat,ural number k snch that k
< p 5 k. Without loss of generality
-
1
are indcpendent. I n the real
m
we may suppose that
2 [i
= k. Then,
using
1
i'k
- 5 exp k! -
(C), k ! 5 k k ,
k1k2 < (1 ip)". , and
= llzllz,
we obtain the desired inequality
< - 2 exp (1)kl'zkl'z 5 cO(l c p)l'z 1 1 ~ 1 , ~ . The complex case
CBn
be treated by decomposition into real and imaginary parts.
(li) q 10.4.3. Lcmina. Let zh = (Cih)', ..., t',) numbers ill,.. ., Asr E (1 - A , -0) such that
l
~
1 ;i.,y1
for h
L
5 c( 1
1, .. ., r , where c
log T ) l i Z sup
n
for
JL
=
I.
..., r. Then.
ljZhj10
h 2s
same positive constmit.
Proof. Suppose that sup /I,zh'i2= 1. Put p := log r . By 10.4.2 we have h
ifrere
me
10. Banaeh Spaces with Approximation Property
for h = 1,
137
. ..,r . Hence
Consequently there exists e, E bmsuch that
rhen R, := fl(e,),
. .., i.,
:=fm(eo) satisfy the desircd condition.
10.4.9. 1,cmma 1. The 3 . 2*-th roots of unity can be decomposed into disjoint subsets (aaJ :j
such that
/I. ,"+l
1,
:
..., 29z+1)and {Bnj : j
I
&I:;] - 2 1'&, 5 cl(n + 1)1'2
j= 1
) =2"1
for h = 1, ..., 3
= 1,
..., 2")
2n12
an,where c1 i s Some poSitiue constant.
-
Proof. P u t 0 := 213, m := 3 2n, and r :I= 3 2". Furtherimre, let
znh := (i.;,,
..., c",)
for h = 1, ..., r ,
= 1,
.... 3
*
P).
can he consiclered a s
(I >: I)-matrices. By 10.4.4
140
Part 2. Theory of Quasi-Normed Operator Ideals
where c is 2 positivc constant. Consequently the ( I inable operator X E B(Zl(I), 7w(I)).Since ( m k nk
=
x
I)-matrix X defines a n approxi-
+ 1' F n
it follows that
Henccl the diagonal (Ix I)-matrix8 represents a nuclear operator S E %(Zm(I), Zl(I)). Therefore N := XX E %(Zl(I), Zl(I)).Moreover, l n k nk ==
I
113 for 0 for
7% = 0 ,
>0
1%
implies trace ( N ) = A-
b,k n k f n k nk =
-
1
I
This proves the assertion, since &(I)and I, coincide.
Remark. More precisely, we have N E %fr,l,lj(Zl, ZI) for
Y
> 2/3; cf. 18.1.1.
10.4.6. Now i t follows the P r o p o s i t i o n . The operator ideal 8 i s neither injective nor surjective. P r o o f . Consider the operator N = S X constructed a t the preceding point. We have X E @(Zl, Zm) and S E %(Zw, ZI). Moreover, S has the form S ( f n )= (antn)with crn > 0. Take t h e canonical factorization
__
where Eo := Z,/N(X) and Fo := H ( X ) .By setting (A,, So):= trace (SJA,&) for
A , E B(&, F,)
we define a functional Sosuch that ! 0, we choose L E S ( H , F1"J) such that rank ( L )< n and lJFS - Lil 5 (1 + E ) ~ ~ ( 8Let ) . P c B ( H , H ) be the orthogonal projection with X ( P ) =- N ( L ) . Then mnk ( P )< ) I , BS well. Moreover, it follows from
/IS - SPil
= 1
iJFS(Tn-- P)ii = /'(JFS - L ) (In - P)li 5 ~IJFS - Lij
that a,(&) I 11s - S P 1 5 1IJp.Y
-
LII 5 (1 + E ) C , & ( S ) .
This completes the proof, since c,(S) 5 cr,(S) is trivial. 11.5.3. P r o p o s i t i o n . Suplmse that P has the metric extension property. Tho! c,,(S)= a,(S) for all S E B ( E , F ) .
Proof. There exists B E l ! ( P j , F ) such that BJF = I , and C,,(S')
I a,(#)
= n,(BJ,S)
$l] 2 1 , Hence
5 /p/I cc,(Jps) ic,(S).
11.5.4. Bn s-functior? s is called injective if s,,(S) = s,(JFX)for all S E D(E,p).
11.6.8. We now state the main result of this section. T h e o r e m . The map c: s -+ L;.
(Cn(S))
the Inyest injectice s-junction.
P r o o f . The properties (OS,) and (OS,) follow imaiediately from the definitionJloreover, (08,)can be checked with the same technique as used in 4.6.2. Condition (08,) is also trivial. Finally, we obtain (OS,) from ~ ~ ( 1 =,1)and 11.5.2. sin^ Pl'" has the metric extension property, by 11.5.3 we have en(& = n,(JFS) = C , ( J d s I ) Let s be any injective s-function. Then
sn(S)= s,(JpS)
a,(JpS) = c,(S) for all S E f?(E,F ) .
11.5.6. We will denote by cod ( M ) the codimension of a given subspace M of h'.
T h e o r e m . Let 8 E 2(B,3'). Then c,(S) = inf { ] ~ A Y J Lc/w~d: ( M I < n ) .
150
Part 3. Theory of Sequence Ideals
Proof. If M isany subspace of E with cod ( M ) < n, then there exists T E 2(X7F i n j ) such that T J g = J F S J z and IlTll = IlSJEII. By setting L := J,S - T we obtain an operator L E g ( E , P n j ) with rank (L) < n. Therefore cn(S) = an(JFS)5 IIJFS - LII = llVl = IlflJEII.
This proves that
.
c,(S) Iinf (iiSJ$I1:cod (a) < n)
To check the equality, given E > 0, we choose L E $(E, Finj) such that rank (L) < n and llJFS - Lll 5 (1 8) c,(X). Put M := N ( L ) .Then cod ( M ) < n and
+
IlflJgI!= IIJFXJgII = II(JFS- L) J,cll 5 IIJFS - Ljj 2 (1
+
E)
cn(S).
11.5.7. Lemma. I{ dim ( E ) 2 n, then c,(IB) = 1. ~
Proof. Let M be any subspace of E with cod ( M ) < n. Then M ~ =I IlJfJJ = ~1. This ~ proves ~ that c,(IE) = 1.
+
(0).Hence
11.5.8. Proposition. Let F be an intermediate q a c e of {Fo.F,} possessing J-type 0. I f S E S(E,P A ) then ,
C~~+,,-1 ~ (4 R F ) 5 C,~(X:E + Fo)'-ec,,(X: E -+ Proof. Given E cod ( i k f k ) < nk and
> 0, according to 11.5.6 we choose subspaccs Mk such that Fkll 5 (1 f &)
I/SJgn:.&?k
Cn,(S:
i!# -+ p k )
for
k
= 0,1
-
+
Put M := iM, n iMl. Then cod ( M ) < no n1 - 1. Moreover, it follows from //SJ$:M -+Fk//5 IISJ&: i&fk -+ $',(I! and the estimate (J) given in C.5.6 that cffo+n,-l(S: E --f F ) 5 IlSJg: M +-PI[
5 lISJZ&:Mo -+ 3',,!11-oIISJZl: dl, + Fllie 5 (1 + E ) cn,(S:E -+ E'0)l-O c,,(S: E -+ Fly. 11.6.
Holmogorov Numbers
11.6.1. For every operator S E 2(E,F ) the n-th Kolmogorov number is defined by := a,(flQL.).
11.6.2. First we prove the Proposition. Let H be a Hilbert space and X E 2(E,H ) . T h r i )
&(AS)= a,(S) = inf { IjX - PSI1 : rank ( P )< n } , where t h infimum is taken over a.11 adm.issible orthogoml p o j e c t i i m s P P r o o f . Given - L[l 5 (1
llXQB
F
> 0,
we choose L E g ( E s ~ rH, ) such tltat rank
+ E) d,(S).
B(H, H ) .
(L)< n
and
Let P E 2(B, H ) be the orthogonal projection with
11. s-Piumbers of Operators in Banach Spaces
152
M ( P ) == M ( L ) . Then rank ( P )< n, a 8 -,yc41. Jlorcover, it follows froill /IS - PSI1 = Il(1, - ) ' 1 S'QCll = Il(1,
-
P)( X Q E
- L)I, d IISQE - LII
that an(S)5 1 1 8-
5 Il8& - Ll1 5 (1
+
E)
dn(S).
This coinpletes the proof. since d,(S) 5 a,@) is trivial.
11.6.3. P r o p o s i t i o n . Suppose that E has the metric liftingp~operty.Then d,(8)=.I.,(&) S E B(E, F ) .
/or ull
Proof. Given E > 0, there exists x'E B(E, E m ) such that QES= 1, anti 2 1 E . Hence
I XI1
+
d,(S) 2 u,(S) = an(A9QEX)5 U , ( S Q E ) IlXl] 5 (1
-+
F)
d,(9).
11.8.4. An s-function s is called sarjective ii S,~(S) = s,(SQE)for a,ll S 5 C ( E , 1'). 11.6.5. We now state the main result of this section. The proof is analogous to that of 11.5.5. Theorem. l'he map
d : s4-(d,(S)) i3
the largest surjective s-function.
11.6.6. We will denote by dim (A') the diriiensioii of a given subspace A' of P. Theorem. Let S E C ( E , F ) . Then d,[S) = inf {!lQ;S11: dim (N) < n ) .
Proof. If W is any subspace of F with dim ( N ) < n and E > 0, then there exists T ~ ( E S FU )~ such , that QPT = Q$SQF and IlTll 5 (1 + E ) '[QiSll. By setting I, := SQE - T we obtain an operator L E C , ( E S U r , F ) with rank (L)< n. Thcrefo1.c d,(S)
= u,(SQE)
5 IlSQE - LI
=
:i!I!'Ii
5 (I + I ) ilQ.<Si'.
This proves that
d,(S) 5 inf {liQ.;SlI:
dim ( W ) < n ) .
To check thc equality, given E > 0, we choose L E a(Es"r, F ) such that rank (1,) llisQ3 - L' 5 ( 1 + F ) dn(S),P u t N :: X ( L ) .Them diin (AT)< n and
< 12
Ciiid
liQ-$Si= j Q$SQ,) = IIQf(SQE - L)lI
5 llSQE - L'g 5 (1
+-
F)
dn(S).
Rcnia rk. Obviously d,(S) is the 7 2 - t h Knlimogorov clianwter (width) of S(YE); d.[LOR, pp. 13%- 1491 acd [PIE, pp. 144--1401. 11.6.7. ;Ai.nin~a. If dim ( E )
n, then d,'(TE)
I.
:
P r o o f . Let N be any subspace of E with dim (A7) < n . Then N uQiIRll = IlQill = 1 by E.1.1. This prorcs that &(Ie) = 1.
+=
B. Kence
152
P a r t 3. Thcorv of Serriiciire Ideals
11.6.8. The following statement is aiialogous t o 11.5.8. P r o p o s i t i o n . Let E be an ilztermediate space of {E,, El) possessing K-type 8. If S E 2(E,, F ) , theft.
1I .7.
Symmetric s-Kumbers
11.7.1. An s-function s is called sy7nmetric if &,(AS)2 sn(S’)for all S 5 P(E, F ) . In case sn(S)= sn(S‘) the s-function is said to he completely symmetric. An s-function s is callcd regulur if sn(S)=- s,&(KFS) for all S E 2(Z,E). 11.7.2. P r o p o s i t i o n . A n s-furrctior, is complelehy symmetric if c1n11 o?&y if it is regular and symmetric. P r o o f . U w the method of 4.3.7.
11.7.3. P r o p o s i t ion. The cxpproximtrfionm~nbersare syrnirietric, but rtot regular. ProoP. Let S E B(E,F ) and E > 0. We choose L E g ( E , F ) such that rank (L)< I L and /IS- LII 5 (1 E ) o n ( S ) . Then rank (L’)< ?L and 1 18’ - L’II 5 (1 c) u,(S). Therefore cr,(9’) 5 (1 t E ) a,(&’). This pioveq that a,(&”) 5 n,(S).
+
+
Tn 11.11.9 and 11.11.10 me will show that / r n ( I : 7, + co) = 1 m d ~ , ~ (ll-+Zm)= I: 112 71 .=. 1 , where I denotes the embedding map froin 1, into co and I, respectiwly. 80 the approximation numbers cannot be regular.
for
11.7.4. We now give a n important suppleinent of the preceding result which states the coniplete spinmetry of the approximation numbers on the ideal of compact operators. Theorem. Let S
R(E, P).Then u,(S)
= un(8’).
Proof. Given E > 0, we choose L E $(E”, 5”‘) such that rank (L)< n auld jlS’ -- Llj 5 u n ( S ’ )+ F. Furthermore, there are yl, ..., ymE F with m
S( ETE) g
u {yz+
&F].
1
..
Let M be a finite dimerisioid subspace of F” containing M ( L ) and KFyl, .,Kf-q,,t. Then, by E.3.1, we can find J , E 2 ( N ,P)such that llJoll5 1 P and li,Joy” = y’’ whenever y“ A’(K17)n M . I n particular, JoKFy2= yi for i = 1, ..., 911. By setting Lox :-- J,Lk-,z for x E E we now define an operator Lo E $(E,P)with < 72, Let x E U,. Then there exists 1 ~ 6such that IISx - yili S F . Henca rank (Lo)
+
j;Sz - L0s1l
5
11S.c- ?/$
+ llyl - .&.? 5 + IIJOKFy, - JoLIiTEzil 8
-&
+ (1 +
E)
5E 5 E.
+ (I +
E ) [F
5 c:
+ ( 1 + ).
+ (1 + ).
alKpy, - ~KEXII [ilK*Wz - KFSXl1
la,(,”)
+ IlKfiSX
+ l/ij””KEZ - LK a ~ l i ]
+ 2&1.
- /.li‘Ezlll
11. s-h'nmbers of Operators in Rannch Spaces
We get
a,(~5 )
(1s- ~ ~ 5l (1 i + E ) (a,(S")
+ 28) +
153
E.
CI,~(S") 5 u,(S') 5 n,(S). This proves that u,(S') = a,@) for
Consequently a,(&) all s E R(E, P).
11.7.5. P r o p o s i t i o n . 1'Ae Gelfund numbere arc regular, but the Kolmogorov numbers are not. Proof. The ro@arity of the Gelfand numbers follows froin their injectivity; cf. &o 11.5.6. 0 1 1 the oth(.r hand, the embedding map from 1, into co shows that the Kolmogorov numb-rs are not. regular; cf. 11.11.9 and 11.11.10. 11.7.6. P r o p o s i t i o n . Let S E O(E, 3').Then c,(S) = d,(S')
a,(&) 2 C,(S').
ft?d
Proof. 8inct. Ji, is a metric surjection, the surjectivitv of the Kolniogorov numbers iiiiplies d,(S') = d,(S'J;.) 5 U,(S'J&) 5 U,(JFS) = C,(S). (1) Since Q i is a itirtric injection, the injectivity of the Gelfand nurrihcrs yields
C,(S')= C * ( Q ; S )
5 fl,'(Qki?) 2 U,(S&e) = d,(S).
(2)
Gsing 11.7.5 and (2) we obtain c,(S) = C,(sr,S)
I L.7.7. Wo
I ~ O Wgive
= C,(S"K,)
5 cn(s") 5 d,(LY).
(3)
a n iliiportmt supplenient of the preceding result
Theorem. Let S E &(E, F). Then c,(S) = dE,(S') nnd
(En(#) = ~~(8').
Proof. Observe that (Ear)' has the metric extension property. Hence, by 11.5.3 and 11.7.4, d,(Sf = a,(S&)
= a,(Q;S)
= cn(Qg7) 5 c,(S').
(4)
11.7.8. T h e o r e m . The Hilbert numbers w e completely symmetric.
Proof. Let S E S(E, F ) . If X E @(El, E ) and B E @(F,K ) such that llXll 2nd jlBll 1, t-hen h,(S') 2 u,(X'S'B')
51
= CL,(BSX).
This proves that hn(S') 2 hn(S). On the other hand, given
E
> 0, according t o 11.4.3 we choose Y E g(Z;. F') and
Obviously there is B E e ( F , 2 ; ) such that Y = B'. Moreover, by E.3.2 we can find S f S(lg, E ) with X'S'Y = AS'Y and llXll 5 1 + E . Hence PI, = X ' S ' B . We now
154
Part 3. Theorv of Seauence Ideals
obtain PI, = BSX, as well. Consequently
1
h n ( 8 ' ) = Q = hn(e&J 14-8
I I141 h,(S) IIXII 5 (1 4- &) U S )
so h,(S') 5 h,(S). 11.7.9. Proposition. There exists an s-function whic7~i s injective, .surjectizw, and completely symmetric. Proof. Put s,(S) := n,(JFSQE)for S E B(E, F ) . Since the other s-number properties are evident, we only check (OS,). For this purpose let E be a Banach space with dim (It)2 n. Supposs that L E ~ ( E S W Ein , j ) and rank (I;) < n. Tlieii, given E > 0. 1 by E.1.2 we can find .co E E s ~ rsuch that //zoI/5 1 -& E and IIJEQEzO - yII 2 I + & 1 for all y f N ( L ) .Hence IlJEQE - LI[2 -This proves that s,(IE) 2 1. (1 + e),'
The injectivity and the surjectivity can be checked as in 11.8.5 m d 11.6.5: respectively.
Let K denote the canonical injection from (3")"' Then QF.= JhK. Therefore a,(#')
= c,(h"Qp)
z=
= Zl(
tTF,) into (PW)'>= 11(UFr)''.
c,(S'J;h') 5 C n ( 8 ' J i , ) 5 dn(Jps)= S , ( 8 ) .
This shows the symmetry.
11.8.
Additive s-Numbers
11.8.1. An s-function s is called additive if
+
s?&*+"z-l(sl 8,)5 s,l(S,)
+ snz(S,)
for all
s,,s, E O W , F ) ,
Remark. This condition contains (OS,) as a special case. 11.8.2. Theorem. The approximation numbers. Gelfand numbers, Kolmog~rovnzmbers, a d Hilbert numbers are additive.
Proof. Given 118, - Lkli 5 (1
E
+
anl+n,-l(sl
> 0, we E)
a,,(&)
+
82)
choose L,, L, E g ( E , F ) such that rank (Lk) < n, and for k = 1, 2. Then rank (L, L2)< n, n2 - 1 and
+
+ - (L1 + 4 ) I I 5 llsi - LlI! + 5 (1 + [a,l(Sl)4-%l(&)I.
2 Il(S1
82)
+
1182
- &ll
E)
The additivity of the remaining s-numbers can be derived from that of the approximation numbers.
11.9.
Multiplirative s-kumbers
11.9.1. A n s-function s is called mnultiplicative if .sm+,,(ST) 5 s,(S) s,(T) for all T E O(E, F ) and S E 2 ( F , G ) .
11. $-Numbers of Operators in Banach Spaces
155
R e m a r k . This condition contains (08,) as a special caw.
i 1.9.2. T heo rein. The approximation numbers, Gelfand numbers, and Kolnqoroc numbers are multiplicative. Proof. Given E > 0, we choose L E 5(E,17) and K E S ( F , G ) such that rank(L) 0, thew are X E Q(Eo,E), IjXIl 5 1, and B E @ ( F , F 0 ) , IlBIl 5 1, such thnt zn s,(BSX) 1sn(S) for
rt =
1 + E
Here Eo and Fa arc szcitable Banach spaces.
1, 2,
...
156
Part 3. Theory of Sequence Idea,ls
P u t E0 :== ll(E,) and F, := l,(F,). Then the desired operators can be defined by thc equations
Here J: and Q: arc the canonical limps defined in C.4.1 and equipped with sonie obvious superscripts. U4ng t,B,,SX, = QffBSXJf we have 7 , s,(BSX) 2 s , ( & ~ B S X J2 ~ )z,s,(B,SX,) 2 s,(4
I+&
*
11.10.4. The next result is analogous to 8.7.11.
I'roposit ion. Euery mnxinurl s-function i8 regular.
11.10.6. An s-function s is called ultrastable if s,2((Si)li) 5
liln sn(Si) 11
for every bounded family (Si)of operators SiE Z(Ei. P,) and every ultrafilter
U.
11.1Q.G. The following result is analogous to 8.8.7.
P s o p o s i t io n. Every ztltrastLrble regular s-f uiact ion i s masirrinl. R e m a r k . Let us mention that, conversely, every iriasiiiial s-function is ultrastable and regular; K. D. KURSTENEl].
3 1.10.7. ,4n s-function s is called weakly lower semi-continuous if
sn(S) lim inf s,(S,) v
for every direct,ed family of operators X, 5 f?(E.3')converging to S weak operator topology.
E B(E,F ) in the
11.10.8. The next statement is similar to 8.7.17. P r o p o s i t i o n . Every maximal s-function is weakly lower senti-continuous.
11.10.9. Leinma. Let ( L a )be a bounded famihj of operutors Li E S(Ei,F E ) .Then mnl; (hi) 71. inplies rank ((Li),)2 n. Proof. By B.4.8 the operators Lican be represented in the forin I1
Li
=
(lik
'3v*
k=l
such that $zikll 5 1 and llyikll 5 IILill. Let a, be the functional on (Ei)ndefined b? (E, a k ) := Iiin (xi,u i k ) U
for x = (si)u
11. s-Numbers of Operators in Banach Spaces
157
and write yk := ( Y ~ ~ It ) ~follows . from
that
Therefore
So we have shown that I, :== (L,)uadmits t h e representation n
L=
a&‘1;y k . 1
Hence rank ( A ) 2
71.
Wc :$re now able t o check the Theorem. The approximntion i~uinbersare ultrastuble. Proof. Let (8,) be a bounded family of opcrators 8, t 2(E,,P,). (hven E > 0 choose Li E g ( E , , F , ) such that rank ( L , ) < TZ and llLT2 - L,ll 2 ( 1 E ) a,(S,). I t follows from
+
rie
llLtll 5 IF, -
+ IIJ.~’A 5 (1 f &) 4 8 , )+ lifl,Il
i(2
+
2)
lls81:
that the family (hi)is also bounded. Hence rank ((LJU)< n. Moreover, we have
This completes the proof.
ll.LO.10. As a n immetli~ieconseqwnce of 11.7.3 and 11.10.4 we get the P r o p o s i t i o n . The appro~
9 m . J
...1,
:= ( E l , * * - , E n , 0,
tn, Enll, En, E n + l ,
--a)
:= (tly
..-) := (51,
-
* .7
En) >
-.*,E n , o ,
***).
The choice of the underlying Banach spaces will depend on the specid situation. 11.11.3. Theorem. Let 1 5 u 5
00.
Theit
a,(S I , + I,) = c,(S: I,, -+ 1,) = &(S:1, + I,,) = 0s.
Obviously rank (L) < n. Hence
P r o o f . P u t L := SP,,.
a,(S: I , +)2,
5 /(AS - L: 1, + lull = 0,.
Suppose that a, > 0. Then 8, :=QnSJn is invertible and IIS;': 1: -+Zz11 = a . ' , Therefore, using 11.5.7 we obtain 1 = cn(In:1; -+ 1;) (= c,(S,: 1; -+ 1:) llS;1: 1;
1:z
3
5 cn(S:1,
-->p
0 , ' .
This proves that o,, 5 cn(S:1, -+ 1,)
S an(&1,
+ 1,)
5 an.
The Kolmogorov numbers can be treated by the sane nlethod; cf. 11.6.7.
11.11.4. To prove the next theorem we need some leinmas. with cod ( A ) < n. Then there em& Lemma 1. Let N be a subspace of = 1) 2 112 - n 1. e r= ( E ~ ..., , cn,) E hrsuclt that llelioo = 1 und card ( k :
+
Proof. We fix some extreme point e of U,. Put
11 := (2:Ek = 0 for k E K ] . K := ( k : /&kl = i] and 2 Clearly card ( K ) f dim ( N )= m. Suppose that card (I 0,
it follows that e f 6x c UAh-. So e cannot be an extreme point, which is a contra1. diction. Consequently we have card ( K ) 2 m - 12
+
Lemma 2. Let 1 d then,
z1
0 and IEntll I nlin (IflI, ...,
11. a-Numbers of Operators in Banach Spaces
159
Proof. Put
We are now prepared to check the Theorem. Let 1 5 27 < u
a,(&: I,
--f
I,)
5 00
= cn(S:I ,
-+ lu) = d,(S: I ,
Proof. First we treat the case 1 Hence
We now suppose that
D(&, ...)&n)
G,
and 1/r = l / v
< u < 00.
- 1/u. Then
41,)
=
2'al;
(nm
ll".
P u t L := SP,,. Obviously dini(L) < %.
> 0 for some rn 2 n. Then by setting
...)o,""Em)
:= ( a p E l Y
we dcfine aonc-to-oneoperatorD from 1; onto lz. By Lemma 1 for every subspzeeH of 1: with cod ( N )< n there is e = (cl, ...,8,) E D ( M ) such that llelim = 1 and I, where K := ( k : lckl = 11. P u t S, := Q,SJm anddeiiote the card ( K ) 2 n~ - N injection from M into 1; by J. If x := D-le, then Lemma 2 iinplies
+
160
Part, 3. Theorg of 8eyuence Ideals
Consequently from 11.5.6 we obtain
c,(S: I,,
--f
Z") 2 c,(S,,,:1:;
--f
1
);2
This proves that,
To treat the case u = co w e only need Lemmti 1. The Kolmogorov numbers can be deterniined by dixality; cf. 11.7.6.
11.11.5. L e m m a . Let 1 u n ( I ::1
-+
4v I :u 5 co and
Zy) ==-
(in - n
+
IIr
/or
1)1/7
= 1/r 7~
= 1.
- l/u. Then
.,., /ti .
Proof. By 11.11.3 and 11.1 1.4 the assertion follows from an(I:1: ->
);z
= n,,(P,,,: I,
3
Z,).
11.21.6. In the following we deal with diagonal operators froiii I , into lp. For this purpose some informations ahout orthogonal projections in 1, are required. Lemma. Let (z,)be a scular sequence. Then there exists an n-dime~isionnlo r t h q m l projection P E 2(12, 2,) urifh IIPeilJ,= 3;for i = 1 , 2. ... i f and only if co \7
U
7c?
I
= 1L
0
rtnn
s z, 5 1.
1
Proof. Since every n-dimensional orthogonal projection P can be written as I&
P
=
2-z5
..., z,) is an orthonormal family, it follows that
xk. where (z,.
1 N
'u
1
i-1
2 1.
Clc*arly0 5
n
I1
5 l/Pei/l:= 2' k_1: ifer, =l
xk)j3
/irk!/; = n.
= 1
f
Conversely, let ( x i ) be a scalar sequence satisfying the above conditions. Then the existence of P can be checked by induction. If ?L = 1, then P := x* x := (z,, z., , ..) is the desired projection. We now suppose that the assertion l i a h been proved for s m i e n. Let W
2 ni =:
11
+1
and
I 2 z1 2 n2 2
1
Then there exists a natural number k such t.hat h
h-,l
')
2 x9 < 1 -=, ; n;. 1
I -
1
Pnt
Ixi
otherwise.
-
0.
161
11. 8-Numbers of Operators in Bmach Spaces
0bviously k
00
znii=n,
En;;=1, 1
and O ~ n o i ~ l .
k f l
Therefore we can find a n orthogonal projection P with
llPeillz = 0 for i = 1, ...,k and llPeillz == zOi for i = k
...,
+ 1, ...
Let zo:= (nol, n o k , 0, ...). Since 1 1 ~ ~ 1 1 , = 1 and Pxo = o tc.e see that the operator Po := : Z @ xo P is a n orthogonal projection with llPOeajjp = noi for 1. By setting i = 1,2, Let 0 5 (Y
+
...
u: :=
1
( 1 - a2)1/$ek -&ek
+ cxek+l
for i = k ,
f (1 - a2)l1'ek+1 for i = k f 1 ,
otherwisci
ez
we define a n orthonormal sequence (ui).Since the operator 00
U,:=E
ef @ u:
1
is unitary, it follows that P, := U ~ P o Uniust , be an orthogonal projection. Observe that Pa is the above Po if LY = 0. Clearly we have IIPaeil12 = zoi= ni for i k and i k 1. Moreover, l/Poekll2 = nOkand iiPlekll2 = Zok+l. Since I,P,eklIpdepends continuously on the parameter a and 7dok 2 n k 2 zbl 2 z o k + l , there exists a. with /IPu,ekllz= nk.Then it follows from
+
=+ +
C llPaee~ll~ =n + 1 W
1
that ll~abek+l~12 = 7dk+l. s o Pao is the desired projection.
11.11.7. T h e o r e m . If (q&) E co, theit n,(S: I ,
--f
7,) = sup
)2: &
h = ?L, n
Uk
Proof. By 11.6.2 we have
i
+ 1, ... .
an(S:I , + I,) = inf { 1 1 8 - PSI/:dim (P)< n ), where the infimum is taken over all aciinissible orthogonal projections. I n view of : S - PSll = sup Il(8 - PS) e&., the preceding lemiria iniplies
11 Pictsch. Operator
162
Part 3. Theory of Sequcnce Ideals
Otherwise let
oh,,
:= 0. It follows from
sup (1 - ,f)l’2 u p i 1 2 (1 - ?
C p
i
that h
h
i
(1 - 3 : ) 2 h -n
1
sup (1 - z;)o ; 2 ’ o i 2 1
-+ 1.
1
Consequently
sup ( I - n 5 ) 1 / 2 oi 2 ohn for 72, = n, n.
+ I , ...
i
This proves that an(&El -> Z2) 2 sup (ahn: h = n, n
+ 1, ...).
Obviously
Hence 0
5 lini sup ohn
oi
ahon= SUP (ohn:h = n, n
ho-n ha-1
2 oi2
and therefore lim ohn = 0. so we can find ho 2 n with h
h
+ 1, ...]. By an easy coinputation, it follows from
s h,-n+l
2
oi2
1
1
and
h,-n+
1
h.
5-
Y
oi2
1
thk
1
(1 - ,;)1’2
Cri
=
ohen
oi
for i = 1, ...,h,, for i = 71, 1,
+ ...,
2 -
h,+ 1 - n + 1 9
he+l
z‘ 1
Oi2
163
11. a-Numbers of Operutors in Banach Spaces
11.11.8. Lemma.
Proof. By 11.11.7 the assertion follows from
un(I:I;" -+ Z?) = a,(P,: I ,
ZZ).
--f
11.11.9. Proposition. U , ~ ( I,I :--f co) = 1 for n = 1, 2, ... Proof. Obviously
an(I:I, -+ co) 5 111:I ,
--f
coil = 1 .
I n order to check a n ( l :I , -+ cg) 2 1 we consider L E g(Zl,co). Then, given there exist y, .,ym E co such that
..
rn
L(u1) S U k/i 1
E
> 0,
+
Here we denote by U , and U , the closed unit ball of I, and,,Z respectively. Write yi = (rib)and choose n with lqinl 2 E for i = 1, ...,m. For some & we have ((Len- yi,llco 5 E. Hence
IiI - L:4
+-
coll L IlIes - Lenll, L Illen - ~ 2 11 - ??i,J - & 2 1 - 2 E .
This proves that 111 - L : I ,
--f
cojj 2 1 and therefore
11.11.10. Proposition. an(l:I , -+2,)
lbi0- Lenlloo
i o ll ~
as(n(lT:I , -+co) 2 1.
= 2,3,
= l / Z for
...
Proof. Put
1
Lo := y x o @ x,, where xo := (1, 1, ...). z
Then rank (Lo)= 1 implies an(I:ZI 3) ,2
5 111 - Lo:I,
->
1,Z
= 1/2.
Suppose that there exists L E @(I1, I,) such that 111 - L: I , + Z,11 denote the E-th coordinate of Lei. We now obtain
- likl5 ] ] l e i- Leill, 5 111 - L: I ,
3
< 112. Let ti,
Z,11.
If i =+ k, then IlLei
- hek]], 2 I&
- tkkl= 11 - iljk
- (1 - &)I 2 1 - 2 [I1- L: I ,
+ Z,1
3 0.
This proves that L( U,) cannot be preconipact, which is a contradiction. So we have 1 1 1- L:I , -+ E,IJ 2 lj2. Hence crn(I:I , -+ E,) 2 1/2.
11.11.11. In the following we use logarithms of base 2. Sublemma. There are xl,...,x, t 2; such that
11*
112i1i2=
1 and
164
Part 3. Theory of Sequence Ideals
.
Proof. If m 5 12, then we can take a n ortlionorinal family (xl,..,xm).Let us now suppose that for soinc m 2 n there are xl, .. ., xmE 1; which have the above property. Put p := log m. If cp denot,es Khintchine's constant, then
+
cp 5 ( [ p / 2 ] 1)112
5 (log m)1/2.
It follows from m
that, for at least one e E P, we have nr
2' \(Xi7 ellp 5 $m1 Put xm+,:= d 2 e . Then
.., rn. So the assertion is proved by induction.
for i = 1, .
We are now able to check the important
L e m m a.
un(l:1';
ZZ)5 3
4
for n
=
1, ..., m.
.., xm E I; as constructed in the suhlemma. Write x i = (Eil, ..., tin)
Proof. Take xl,. n
P u t L := 2 z ! @ x!, where x! := (tlr, ..., f m f ) . Clearly rank ( L )5 n. Moreover, 1
the operator L is represented by the matrix ((xi,xk)). We now see from
C L ~ + Zy ~ (+ I : Zg) 5 )!I- L : Z';" -+ZZli 5 2
log m
This completes the proof. R e m a r k . For nz 2 2 we have
ul(I:1';" --f l g ) == 1, a2(I:1': + 1:) = 112, and u m ( I :1';
--f
Zg)= l / m .
However, it follows immediately froni 11.11.8 that a&:
1';"
4
I",") 2 cntllz.
Using number-theoretical methods R. S. ISMAGILOV [ 2 ] proved that
11.11.12. It seeins very likely that the preceding lemnia can be extended in the following way. For a somewhat weaker estimate we refer to B. CARL/A.PIETSCH [2].
11. s-"nmbers of Operators in Banach Spaces
C o n j e c t u r e . Let 1 5 u 5 2. Then there exists some constant e
165
> 0 such that
11.11.13. Finally, we mention a very striking result which has been recently proved by B. S. KASHIN[3] and B. S. MITJAGIK[a]. Lemma. There emysts 0 such that
ZF) 5 p
c,,(I:2;" -+
11.12.
[log (7n
+ 1)]3'2
nI/2
for n = 1,
..., m .
Relationships between s-Plumbers
11.12.1. As a consequence of 11.2.3 and 11.4.2 we have the Theorem. Let S E O(E,3').Then US)
s c,(&
d4s)
and
h,(4
5 4 2 ( W 5 a,,(S).
11.12.2. In the followiiig some converse estimates are checked. Theorem. Let S E O(E, F ) . Then a,(S) 5 Zn1/2c,,(S) and a,(&) 5 2dkI,,(S).
Proof. Given F > 0, we chooseanoperator L E g(E*u, F ) withrank ( L )< n and - I,]/ 5 (1 + E ) d,(S). By B.4.9 (remark) there exists a projection P E B(P,P) such that M(P)= M ( L ) and llPil 5 (n - 1)1/2. Obviously rank (P)< n. Moreover, it follows from
IP - PSI1 = iiVF - P ) SQEII= IIUF - P)(SQE- OII 5 (1 + ( n - 1)1/2) llSQE - L!l that a,(#)
p /IS- 15s;;5 (1
+ (n - 1 ) q
IISQE
- LI/ 5 (1
+
E)
2n1'2dn(s).
The remaining estimate can be proved by the same technique. R e m a r k . If 0
< a < 112 and p > 0, then estimates of the form
a,(S) 2 gnv,l(s) or a,(S) 5 ,oned,(X) cannot hold for all S E B(E.F ) ; cf. 11.11.8 and 11.11.13.
11.12.3. Theorem. Let S E ,f?(E,3').Then n
n ck(S)5 1
> 0,
fl hk(8)
n dk(S)5 enn!fl h k ( 8 ) . n
11
18
e%!
and
1
1
1
-&(A').
Then there exists z1E E with I+& ilzll/ 5 1 and IISxlll 2 al. By induction we can find zl,.., x,,E E such that llzkll 5 I and Il&;,Sxkll 2 6, for k = 1,. .., n. Here Nkdenotes the linear span of (19.z~: i < k]. P r o o f . Given
E
we put
BK :=
.
166
Part 3. Theory of Sequence Ideals
Next we choose b: E (PINk)‘with 1lb:ll 5 1and I(&;,Sxk, b;)l 2 dk. If bk := (Q”,)’bE, then 1(Szk,bk)l 2 dk and (Sxj, bk) = 0 for i < k. Hence the matrix ((Sxi,a,) has superdiagonal form, and it follows that n
[det ((8%h))l==
fl I(fl%
n
6k)l
n8k 1
1
By setting
2
n
X(Ei):= 2’ tizi and By := ((y,
a,)
1
we now define operators X E f?(g,E) and B E 2(F,Zi). Clearly llXll 5 d 2and l[Bll 5 n1I2.Put So := BSX and consider a Schmidt representat,ion n
so= 2’ a&
@
Vj
1
such that al 2 ... 2 equality (27.2.3) froin
CT,~
2 0. Then ai = aj(So)5 nhj(S). Using Hadamard’s inn
(Szi, h) = (Soei,4 = 2 (ei, uj) aj(vj, ek) j=1
we get
n 1
r n” n hi(&). n
A
5
of
1
Clearly 12’ 5 enn!.This proves the second estimate. Finally, the first estimate follows by 11.7.6 and 11.7.8.
11.12.4. Problem. Does there exist some constant e
> 0 such that
c,(S) 2 enh,,(S) and dn(S)2 e?zh,(B) for all S E f!(E,P). Remark. Weaker estimates have been proved by W. BAUHARDT [l].
11.13. Notes I n the context of integral operators the concept of s-numbers first appeared in a paper of E. SCHMIDT[I]. The generalization to approximable operators in Hilbert spaces probably snd R. SCHATTEN [l]. An excellent treatise of this theory is goes back to J. VON NEUMANN the monograph [GOH, pp. 24-64]. In Banach spaces there are many different possibilities [l] has introduced the to define some equivalents of 8-numbers. At first A. N. KOLMOGOROV so-called n-th diameters (widths) of bounded subsets, and then I. M. GELFANDsuggested that there is also a dual concept. The approximation numbers have been studied by A. PIETSCII [2]. The reader is warned that the axiomatic theory of s-numbers presented in this chapter differs from that developed by the authors in [18]. There is an extensive literature devoted to computations and estimations of 8-numbers of diagonal operators in sequence spaces and of embedding maps in function spaces.
11. s-Numbers of Operators in Bmacli Spaces
167
flccommendrttions for further reading:
[DUN, [LOR], [PIE], [SAT,], [SENl, exp. 18, 191, [SE-,
exp. 171.
W.BAUHARDT [l],N.S. BIRMAB/N. Z. SOLOMJAK [l],B. CARL/A.PIETSCE (21, M. FIEDLER/ V. P T a K [I], E. D. GLUSKIN[l], I. C. GOHBERG/M. G. KREJN[l], R. A. GOLDSTINE/R. SEAKS 111, C. W. HA [l], C. V. Humox [l], C. V. HUTTON/J. R. MORBELL/J. R. RETHERBORD [I], R. J. ISMAGILOV [l], [2], P. D. JOIINSOB [l], B. S. KASHIN[l],[ 2 ] , [3], H. KONIG[a], K. D. [CURSTEN [I], G. G. LORENTZ [l], V. E. MAJOROV [l], A. S. NARKCS [l], B. S. MITJAGIN [l], i4], B. s. MITJAQIN/G.31. CHENKIN [I], B.s. &fITJAGIN/A. PEECZYP~SKI [I], B. 8. MITJACIIN/ V. $1. TICHOMIROV [l], A. PIETSCH[20], 8. J. ROTFELD[i], M. Z. SOLONJ.AK/V. M.TICHO.~IIP.OV[l],V. M. TICHOJI~KOV [l], [2], H. TRIEBEL[l], [2], [3], [4].
12.
Entropy Numbers of Operators in Banach Spaces
I n the folloaing we introduce the so-called entropy numbers of operators in Banach spaces. This concept is more suitable for generating operat,or ideals than the c-entropy defined by L. S. PONTRJAGIN and L. G. SCHNIRELMANN in 1932. First we state the basic properties of entropy numbers. Then, as an elementary example, diagonal operators in classical sequence spaces are investigated. Finally, we deal with the relationships bet,weenentropy numbers and some special 8-numbers. In what follows all logarithms are to the base 2. For simplicity we consider real Banach spaces only.
12.1.
Outer and Inner Entropy Numbers
12.1.1. A map s which assigns to every operator S a unique sequence (sn(S)) is called a p8eudo-s-function if the following conditions are satisfied: (OS,) liSll = sl(S)2 sz(S)2 ... 2 0 for S E B(E,P). (08,) S n ( S T)5 s,(s) IlTll for 8, T E B(E,P). (OS,) s,(RST) 5 IlRll s,(S) IlTII for T E B(Eo, E ) , S E B(E,F),R E B(F, Po)Concepts such as injectivity, surjectivity, additivity, multiplicativity, and maximality can be carried over from s-functions to this more general situation.
+
+
12.1.2. For every operator S E B(E,F ) the n-th outer entropy number en(S)is defined to be the infimum of all o 2 0 such that there are yl, ..., yq E F with q 5 2*-l and 4
fwd s u
+
{?A
1
O77FI.
12.1.3. Theorem. The map e: S -+ (en(#))
i s a pseudo-s-function. Proof. First we check IlSll = e,(S). It follows from
lSl1 := inf {o 2 0: S(UE)G C u p ] that el(#) 5 $Sil. We now assume that S(U,) C= yo + aU, for some yo E F . If x E UE, then there are y+, y- E UF with +Sx = yo + ay, and -Sx = yo + q-. Hence 2 IlSxll = c j/y+- y-11 5 20. So we have l/SIl2 o and therefore IISll 5 el@). This proves property (OS,). The remaining conditions (OS,) and (08,) are special cases of 12.1.4 and 12.1.6.
12.1.4. Theorem. The outer entropy numbers are additive. Proof. Let that
s,,8,E g ( E ,F ) . If
sk(UE)
‘t
5
ak
> e,,(sk), then there are yik)’,.. ., yb: E F such
4k
u {y:k’+ O k u F ) i=l
and qk
for k = I, 2.
12. Entropy Numbers of Operators in Banach Spaces
169
u,, we can find i k with s k x E ?,Jif’ f (TkuJ?for k = 1, 2 .
Hence, given x E
This implies
(81f
82)
x t ?At’ f y:f) f (el
+
62)
K’F
and therefore v,
+ S,) ( U c ) s u
Pa
u iL=l
(81
{ykf’
i,-1
+ Ylf’ + + c?.)U p ) . (61
Xoreover, qlqe 5 2(nl+n*-1)-1. Hence en,+n,--l(S1+ 8.J 5 el the proof.
+ cz. This completes
12.1.6. Analogously we obtain the T h e o r e m . The outer entropy numbers are multiplieutive. 12.1.6. For every operator S E P(E, F ) the n-th inner entropy number fn(S)is defined to be the supremum of all e 2 0 such that there are xl, ..., xp E with p > 20f-’and
rJE
k. ] / S x i - Sxkll > 2e for i =i= 12.1.7. The next result can be established easily. Theorem. The map
f: 8
+
( f n m )
i s a p8eudo-s-fumtion.
12.1.8. The following statement is also trivial. P r o p o s i t i o n . The inner entropy numbers are injective and surjective. 12.1.9. P r o p o s i t i o n . The inner entropy numbers are maximal. Proof. Let e < fn(S).Then we can find xl. ...,x,.,E U, such that p > 2R-1and / I ~ z ,- Sx,ll > 2e for i + k. Choose b1k E UF,with l(b’xz - s x k , bzk)l = \lsxA- SX,]~. Let M be the linear span of xl, ..., xp and put
N := {2/ E F : (y, bL,:)= 0 for i, E
= 1,
...:p } .
Then
liQ$SJ$x, - Q$XJE,x,ll 2 I(Sx, - Sxk. a,,)]
> 29
for i -j= k.
Hence fi,(S) = sup {fn(Qf8J,”,):ill t Dim ( E ) ,N E Cod ( F ) } .
12.1.10. We now compare outer and inner entropy numbers. Theorem. Let S t B(E,3’).Then fn(S)5 en(@ 5 2fn(S). P r o o f . Suppose that :md yl,..., y4 E F with
6
IjSx, - Sx71j> 20
> e, Zn-l 2 q. So there must exist different elements Sx,and Ssi which belong oU,. Consequently 2~ < j\Szi- SzJ 5 20. This proves that to the same set 7jk f n ( S ) 2 e,(Q Given e > f,&(S), we choose a maximal family of elements rl, ..., xp E UE such that l\Ssi - Sxk\\> 2e for i f E. Clearly p 5 2-1. Moreover, for L U Ewe can find borne i with jlSx - Sxi/l 5 2p. This means that
+
P X(uE)
!G U {Ssj 1
+ ~QU,).
So %(AS’) 5 2~ and therefore e,(S) 5 2fn(S). I2.1.11. P r o p o s i t i o n . Let E be an intermediate space of {Eo,E l f possessing K-type 0 I f S € 2( EZ,F ) , then G.+~,-~(S: E -+ F ) 5 2enO(S: Eo + F)l-e e,,(S: El
3
F)O
p r o o f . Suppose that crk > e n r ( 8 : Ek .+F ) . Then we can find pk
S(uE,) 5 u (!/‘,“’ h=l
+
+
+ (1 + e ) := (1 +
s u u kJoyIp.’ + ely;:’
+ 2(1 +
ho=l
E F with
be given. By C.5.5 there are 1-8 R c1 I/zljll5 (1 E ) c0 cl.
+
+
0:-’0TuF
Hence
Qa
S(U,)
>0
oo llzollo
u;~cTA-‘IY~.
for some hk, where @k
....’:y
GkuF)
m d qh 5 28e-1 for k = 0, 1. Let 2: E UE and E zo E Eo and x1 E E, with ro x1 = z and Obviously we have
Sxk 6 @kyLt’
ZJ;’),
E)
91
hl=l
S o w it follows from qoql 2
E)
o;-60!UF1.
that
ea,+nl-l(S:E -+ P ) I 20;-eu!. This completes the proof.
12.1.12. P r o p o s i t i o n . Let F be an intermedinte space of {F0,PI}possessing J-t?jpe 6 . I f S E B(E, F A ) ,then en,+nl-l(S:E
--f
Proof. Suppose that with
F ) 5 2ens(S:E -+ Po)‘-’ enl(S: E + F,)’. crk
c= u {Yp’ +
> e,,(S: E .+Fk). Then
we can find y:”,
..., ?/it’E Fn.
4k
S(U,)
0kUF,I
h=l
and Put
k
t&
5 2mk-1 for k == 0,1, Let xI,..., zpE I , := (i:8x6 E yp’
+ a0Upo}.
uEbe
given, where p
> 2(”0in1-1’-1.
12. Entropy Kumbers of Operators in Banach Spaces Ps
card (I,)2 p
Since
171
> yoql, we have card (I,,) > q1 for some ho. Hence there are
&= 1
i, j E Ih, such that Sx,and Sxi belong to the mme set yit' -+ glUF,.This mcans that ' / S X~ Sxl//k5 2ck for k = 0, 1. Now C.5.7 yields jlSx, - SXJi 5 2o;-'ei. Hence fnoL,z,-l(S E: -> F ) 5 IT;-'CT?.
So, by 12.1.10, we obtain the dvsired estimate. 12.1.13. Finally, it is shown that the outer and inner entropy nnnlbers do not have all s-number properties. P r o p o s i t i o n . If dim ( E ) = m ,then
Proof. Observe that there is a translation-invariant Bore1 measure p on E . Obviously we may assume that p ( U E )= 1. Then p(OUE)= em for all c 2 0. Let P
UE
U [xi
+ oUE)
and q
5 P-1.
1
it follows from Q
1 =p(uE) 5
/.&(xi
0 u B ) = qcm
1
that u 2 q - 1 ' ~ 2 2-(n-l)/m. So the first estimate is proved.
+
e)/e = 2(*l)lVn.Since 77, is compact, we can find a Let e be determined by (1 maximal family of elements zl, .... xp g UE such that IlSzi - SZ~II> 2e for i k. Then the sets x i eUE are disjoint and we have xi & (1 f @)uE. Hence
+
13.2.
+ ,puE
Entropy Numbers of Diagonal Operators
12.2.1. First we deal with identity maps between finite dimensional spaces. Lemma. Let 1 5 u, v 5 co. Then
+-
172
Part 3. Theory of Sequence Ideals
P r o o f . Let us denote by Suppose that 9
L7:
arid U y the closed unit ball of 1; and l?, respectively.
+ (iUy}
t7: 2 (J (yL
and q
2-l.
1
Then 9
i(UZ)5
Z(yt I
+ UUY) = qo"i.(U;"),
where h is the Lebesgue measure on W71'. Now A(Ug) = 2" and /?(UT)= P / n z ! imply U* 2 m!/2m-1. Using e:%! > m* we get (i > m / 2 e . Therefore
_- Zy) 2 m / 2 e .
e,JI:: 2
Finally, it follows from
Zz
em(I:
--f
Zy) 5 111:1; -+ZTii e n a ( IE':," - m 1 L l ' U - l ' V e m ( I : I?
e ) for 0 < e 5 al. We now show that
Put nz := max { k :c k 2 E ) and 8, := &,SJ,. We denote by U r and U g the closed unit ball of 1r and EZ, respectively. If y t Sm(Up), then there exists a n integer valued vector g = ( y l , ..., y m ) such that y
Sinco el 2
E Ern-llZ(2g
+Uz]
+ EUF
2~rn-1129
... 2 a,,, 2 E , we have
+
~m-92g Uz]
S m ( U r )+ 2~m-112UZ
3Sn1(U.3.
Let (gl, ..., go] be the collection of all gr = ( y i l , ..., y i m ) with
&m-~~Z(2gi -t U E } G 3Sn1(UF). Clearly 9
( , ?!A
ur)& u
(2ET?2-112 gi
1
+ EUrl
and therefore q
+
S ( U 2 )& JmSm(UF) em+lU,
u (2emn-llzJmgi+ ~ E U , ) , 1
where U , denotes the closed unit ball of I,. On the other hand,
+ uz}]5 zrn1b(
q
p[&m-'"lrnA( uz) = 2 I[&m-112[2gi 1
m
U r ) flak
9
1
where I is the Lebesgue measure on z". Using Stirling's forinula we get elI'(t
Hence
+ 1) 2 (2n)1/2tttll2
for O
< t < 00.
12. Entropy Numbers of Operators in Banach Spaces
-.1+q -
P
log ( 1 j t ) d t
P
175
[: ] . o;
0
This completes the proof.
12.3.
Relationships between Entropy Numbers aiid s-Numbers
12.3.1. We begin this section with an easy result. Theorem. Let S g B(E, F ) . Then h,(S) 5 2e,(S). Proof. By 11.4.3, given z.
lIXl1 5 1, IIRll 5 1, and BSX that cn(In)2 1/2. Hence
> 0, there exist X = PI,, where
B(Z!, E ) and B E B(P,1:) such that
1 e =h,(X).
I+&
1 ha(&)= p 5 2e,,(PI,) = 2e,(BSX) 5 2 e , ( ~ ) . I+& This completes the proof.
M7eknow from 12.1.13
176
Part 3. Theory of Sequence Ideals
12.3.2. Theorem. Let S E B(E,F ) . Thew c,(S)
5 ne,(S)
ccnd
d,(S)
5 me,(&).
Proof. If 0 < g < c,(S), then we can inductively find xl,..., x, E U E and ..., b, E up such that l(s.rk, bk)l == IlsZkll > @ and 2, &!k. Here the subspacelVk with cod ( f x k ) < k is defined b y Mk :=(x E E : (sx, b,) = 0 for h < kt. 61,
denote the set of all vectors e = ( E , ,
Let
Obviously card (8,)= 3. If e'
. .., E,) with
q = f l . Then
=+e r r , then we put h := niin ( k : + E;
E;).
Since
it follows that I/S.z,. - Sze.ii > 2e/n. Therefore e,(S) 2 f n ( S )2 p/n. This proves ne,(S). that c,(S) The second estimate can be checked analogously.
12.3.3. T h e o r e m . Let S E S ( E , B). Then
Proof. Given (= (1
//S -- L II
L
X
t
L
+
E
> 0,
E)
= JLoQ be
we choose a n operator L E g ( E . F ) with rank ( L )< m arid
cc,(S). Then
the factorization described in A.3.4. It follows froin 12.1.13 that
This completes the proof. R e m a r k . The case m = 1 is trivial, since e,(S)
5 IISlJ.
12. Entropy Numlicw of Opcrntors in R m a c h Spaces
12.4.
177
Notes
The notJionof &-entropyfirst appeared iri t,he classical paper of L. S. PO'ONTRJACIN~L. G. SCHXII:E:LNIAN [i]. Many results concerning the &-entropyof special subsets in function spaces can be found in A. N. KOLMOGOROV/V. 31.TICHONIIROV [l] and G. G. LORENTZ ti]. We also refer to the monograph [LOR, pp. i5O-i67]. As B. S. MITJAGIN[t] observed in 1961 the &-entropy is n powerful tool to dinracterize nuclear locally convex spaces. I n the same paper he studied the behaviour of this function for ellipsoids in Hilbert spaces. The theory of the so-called entropy numbers has been developed by the aut,hor and is here presented for the first time. However, certain functions inverse to the &-entropy already tbppear in B. S. MITJAOIS/ A.PEEEZY&SEI 111 and H. TRIEBEL[3]. tkcommendations for further rexlitig :
[SEM,, exp. 18-201, [TRI].
R. lr. DUDLEY[I], 31. B. R. OLOBE L3], H. TRIEREL [1].
K.CARL/A.PIETSCH [I], (I'NYIOXIROVEl],
J[tRCUS
[I], B. S.NIT.JIGIN/S'. M.
13.
Sequence Ideals on the Scalar Field
I n this chapter the concept of a sequence ideal on the scalar field is introduced. The most important examples are the classical spaces 1, with 0 < p < co. Roughly speaking, we can say that the theory of sequence ideals is a simple counterpart of the theory of operator ideals. I n the following we only give some basic definitions and formulate many results without proofs. This is motivated by the fact that the analogous statements have been checked in the context of operator ideals. Furthermore, we mainly treat quasinornied sequence ideals. The modifications for the general case are left to the reader. Finally, we describe further examples due to G. G. LORENTZ, W. L. G. SUGEKT and W. ORLICZ.
13.1.
Sequence Ideals
13.1.1. Let us recall that 1denotes the set of all bounded scalar sequences. Moreover, := { 1, 2, ...). we put A sequence ideal a on the scalar field is a subset of 1satisfying t,he following conditions :
dv
(NI,) el E a, where el is the 1-st unit sequence. (NI,) It follows from xl,x2 E a that x, x2 E a. (NI,) If s E 1 and x a, then sz t a. (NI,) Let z be any one-to-one map from .N into itself. Then
+
J,x E a a.nd Q x x E a for all x E a. R e m a r k . Scalar sequence ideals will be denoted by small Gothic letters. R e m a r k . I n place of (XI,) we may also assume that a is permutation-invariant. However, to get analogy with (QNI,) we prefer the above version.
13.1.2. We mention the trivial P r o p o s i t i o n . Every sequence deal i s a linear space.
13.1.3. For later use we define the map D which assigns to every sequence z = ([I, 62, ...) the doubled sequence Dx := (&, &, E,, [,, ...). L e m m a . Let a be a sequence deal. Then x E a implies Dx E a.
D
P r o o f . P u t .;(n):= 2n - 1 and B(n):= 2n. Then the assertion follows froin = J, JB.
+-
13. Sequence Ideals on the Scalar Field
179
13.1.4. A sequence s = (5,) is called finite if it has only a finite number of coordinates tn 0. The set of all finite sequences is denoted by f.
+
13.1.5. T h e o r e m . f i s the smallest sequence ideal. 13.1.6. The set of all zero sequences is denoted by c,. 13.1.7. A sequence ideal a is said to be proper if a
+ 1.
13.1.8. T h e o r e m . c, i s the largest proper sequence ideal. P r o o f . The ideal properties of c, are evident. We now consider a proper sequence ideal a. Let us suppose that a $ c,. Then there exists x c a \ c,. Consequently we can find E > 0 such that i l l := ( n :\tn12 E ) is infinite. Obviously the sequence s = (c,) with cn := 5;' if n c $1 and a, := 0 if n 6 ikt is bounded. Hence efif = sx E a. P u t e = (1, 1, 1, ..-). If n is any one-to-one map from N into itself such that n ( N )= M , then it follows froin (XI,) that e Q,eM E a. So Q = I, which is a contradiction. ---J
13.2.
Qua,si-Kormed Sequence Ideals
13.2.1. Let a be a sequence ideal on the scalar field. A map a from a into 9?+ is called a quasi-norm if the following conditions are satisfied. (QNI,) a(el) = 1, where e, is the 1-st unit sequence. (QNI,) There exists a constant tt 2 1 such that
+
+
a(xl z2) 5 x[a(xl) a(xz)] for xl,x2 E a. (QNI,) If s E 1 and x E a, then a(sz) 5 ( ( ~ ( 1a(%). (QN13) Let z be any one-to-one map from JV into itself. Then a(Jzz) (= a(x) and a(Q,x) 5 a(x) for all
5
E a.
R e m a r k . Quasi-norins on scalar sequence ideals will be denoted by sinall Roman let,ters.
13.2.2. P r o p o s i t i o n . Let a be a sequence ideal with a quasi-norm a. Then a is a linear topological Hausdorff space. 13.2.3. A quasi-normed sequence ideal [a, a] is a sequence ideal a with a quasi-norm a such that a is complete. 13.2.4. P r o p o s i t i o n . Let [a, a] be a quasi-normed sequence ideal. Then 1]x1]5 a(x) E a.
for all x
Proof. It follows from
that 11x11 = sup n
13*
5 afs).
180
Part 3. Theory of Sequenrr Ideals
13.2.5. L e m m a . Let [a, a1 be a quasi-noinzed sequence idctrl. Then a(Dx) 5 2a(x:) jor all .c f a. Proof. Using the notation of 13.1.3 we get a(&)
=
a(J,c + J p ) 5 ?a(%).
13.2.6. A quasi-norm a on the sequence ideal a is said to lw a $1-norm (0 < p if the p-triangle inequality holds: a(zl
If p
+
.zr)p
= 1, then a is
5 1)
5 a(xl)p + a(.x,)p for xl,xz u .
simply called a norm.
13.2.7. A p-wormed sequence ideal [a, a] is a sequence ideal a with a p-norm a such that a is coniplete. If p = 1, then [a, a] is said to be a n o r m d sequence ideal. 13.2.8. The following statement is analogous to 6.2.5. T h e o r e m . Let [a, a] be n quasi-normed sequence idecil. T ~ Pthere ~ Lexists an equivalent p-norm ap.
13.2.9. Theorem. [c,, 13.2.10. Let 0 < p is denoted by I,. we put
\].I\] i s a noriried sequence ideal.
< co. The
set of all absolutely p-summable sequences z = ( f n )
R e m a r k . It is convenient to define [Iw, l,] := [l. li.\i].
13.2.11. Theorem. El,, lP] i s a nonned sequence ideal / o r p 2 1 and a pa-normed sequence ideal for 0 < p < 1. 13.2.12. Let [a, a] and [b, b] he quasi-normed sequence ideals. Then [a, a] means that a b and a(z) 2 b(s) for all z E a. 13.2.13. P r o p o s i t i o n . Let 0 13.2.14. We put I, :=
< p , 5 p , 5 ce. T h e n
ip,]
r-
[h, h]
[fpz, IPJ.
n I,. P>O
13.2.15. P r o p o s i t i o n . 1, is a seqzieilre ideal.
13.3.
Procedures
13.3.1. A rule
new: [a, a] -., [anew, anew] which defines a iiew quasi-normed sequence ideal [anew, anew] for every quasi-normed sequence ideal [a, a] is called a procedure.
13. Sequence Ideals on the Scalar Field
R e m a r k . The quasi-nortned sequcncc ideal as [a, a]new.
13.3.2. We now list
seine
[anew.
181
anew] will frequently be writtcii
special properties :
(31) If [a, a] & [6,b], then [a, a]new C, [b, b]new (monotony). (I) ([a, a]new)new = [a, a]new for all [a, a ] (idempotence).
A monotone and idempotent procedure is a hull procedure if [a, a] & [a, aInew and a kernel procedure if [a, a ] 2 [a, a]new for all quasi-nornied sequence ideals. respectively.
13.4.
Ninimal Quasi-Xormed Sequence Ideals
13.4.1- Let [a, a] bo a quasi-nornied sequence ideal. A sequence x belongs to the ?iLi?zimd kernel a m if x = tx, with t E co and x, E a. We put amin(x)
..-
inf llltll a(J-ll)I,
where the infinium is taken over all possible factorizations.
13.4.2. The following statement is similar to that of 8.6.2. T h e o r e in. [amin, amin] .is a. quasi-normed sequence ideal.
13.4.3. Moreover, we have the Theorem. The rule
ntin: [a, a] + [amin, amin] .is a h n e l procedure.
13.4.4. For every sequence x = (En)we set
Analogous to 10.3.3 we now prove the Theorem. Let [a, a ] be a quasi-normed sequmce ideal. Then amin consists of those sequences x E a such that
x = a-lim Pnx. n
Proof. First observe that a m h ( z ) = a(x) for all x E f. If x E a m h , we can find a factorization x = txo, where f E co and zo E a. Then a(x - P,lx)d !It - P,tll a(ro). Consequently x = a-lim Pnx. n
182
Part 3. Theory of Sequence ldeals
We now consider a sequence x E a with x = a-limP,x. Then (P,x) is a n n
a-Cauchy sequence. It follows from amin(P,x - P,x) = a(P,z - P,x) and 13.4.2 that (Pnx)converges in a m h . Since x is the only possible limit, we obtain x E ami". R e m a r k . The above statement means that amin is the a-closure of f in a.
13.4.5. A quasi-nornied sequence ideal [a, a] is called minimal if [a, a] = [a,
&Imin.
13.4.6. We now give the main examples. P r o p o s i t i o n . T h e quasi-normed sequence ideals [lp,lp] with 0
[c,, 11.111 are minimal. 13.6.
< p < co and
Maximal Quasi-Normed Sequence Ideals
13.5.1. Let [a, a] be a quasi-normed sequence ideal. A sequence z belongs to the maxim1 hull amax if tx E a for all t E c.,
We put amax(x) := sup (a(tx):t E c, and lltil 5 1 ) .
13.5.2. The proof of the following statement is similar to that of 8.7.2. T h e o r e m . [amax, a""]
i s a quasi-normed sequence ideal.
13.5.3. Moreover, we have the T h e o r e m . T h e rule
max : [a, a] --f [amax, amax] i s a hull procedure.
13.5.4. Analogous to 8.7.5 we now prove the T h e o r e m . Let [a,a] be n p-normed sequence ideal. T h e n sequemes x E I such that (P,x) is a-bounded. I n this m e ,
amax
co?zsist8 of those
amax(x) = sup a(P,x). n
P r o o f . If z E
amax,
then we have a(P,x) 5 amax(.).
Conversely, suppose that a(P,x) 2 a(Pn(tz)- P,(tx)) 2 /jP,t
-
e for n = 1, 2, ... Let t E c,.Then P,tll a(P,x) 5 I'P,t - P,,,til Q for n > m.
Hence (P,(tx)j is an a-Cauchy sequence endtherefore tz ;a. This proves that x E amax. Moreover,
-
a ( t z ) = lim a(P,(tz-)) = sup a(P,(tx)) 5 IItII sup .(Pnz) n
This proves that
amax(z) =
n
sup a(P,.r). n
n
13. Sequence Ideals on the Scalar Field
183
13.5.5. A quasi-nornied sequence ideal [a, a] is called maximal if [a, a] = [a, a1-X.
13.5.6. We now give the main examples. P r o p o s i t i o n . The quasi-normed sequence ideals [I,, l,] with 0 niaximal.
< p 5 co w e
13.6.7. Let a be a quasi-norm on the sequencc ideal a. Then a is said to bc weakly lower semi-continuous if a(.-)
2 lim inf a(x,)
for every directed family of sequences x, E a coordinatewise converging to x E a.
13.5.8. As a counterpart of 8.7.17 we have the P r o p o s i t i o n . Let [a, a] be a maximal p-normed sequence ideul. Then a i s weakly louer semi-continuous.
13.6.
Adjoint Kormed Sequence Ideals
13.6.1. Let [a, a] be a normed sequence ideal. A scquence .- = (6,) belongs to the ndjoint sequence ideal a* if there exists a constant o 2 0 such that
1:2’ I
lnvn 5 as(y) for all 1~ = (vn)E a.
We put
a*(x) := inf o . R e m a r k . The adjoint normed sequence ideal [a*, a*] will frequently be written as [a, a]*.
13.6.2. The following statement is analogous to 9.1.3. Theorem. [a*, a*] i s a normed sequence ideal.
13.6.3. The next result is evident. P r o p o s i t i o n . Let [a, a] and [b, b] be normed sequence ideals. Then [a, a] & [6, h] implies [a, a]* 2 [b, b]*.
13.6.4. Moreover, we have the P r o p o s i t i o n . Let [a, a] be a normed sequence ideal. Then [a, a] E [a, a]**.
13.6.5. The main exaniples are given in the following P r o p o s i t i o n . Let 1 5 p 5 co. Then [l;, If] = [Ipt, I,*].
13.6.6. A nornied sequence ideal [a, a] is called perfect if [a, a]
=
[a, a]**.
13.6.7. P r o p o s i t i o n . The normed sequence idenl [lp, I,] i~ perfect for 1 5 p 4 ob. 13.6.8. A normed sequence ideal [a, a] is called self-adjoint if [a, a] = [a, a]*.
184
Part 3. Theory of S r q i i c n w Ideals
13.6.9. Theorein. [I,, l,] is the mcly self-trdjoint normed sepirnrt’ ideal. Proof. By 1 3 . 6 3 we have [I,, 1,]*
[I,, 12].
: :
Let us now suppose that the norrnod sequence ideal [a, a] is self-ndjoint. Then it follows that
>;
u2
W
IEnl~=
En& F a(%) a(.?-)== n(.c)2
1
Hence x
for
5
a.
1
1, and 12(z)5 a(%).This proves that [a. a ] E [I,, I,]. By 13.6.3 we obtain
[a, a] = [(I, n]* 2 [I,, l,]* Consequently [a, a] = [I,,
-=
[12, 1J.
41.
13.6.10. FinalIy, we formulate
B
result whicli is analogoils to 10.3.8.
Theorem. Let [(I, a] be a minilnnl normed sequence ideal. I f y
a*, the72
00
(x,
Y> :=
s
SnTn
1
defines a functional y on a. The correspondence obtained in t h i y w a y i~a metric Gomorphimu between [a, a]* arLd
ra, 4‘.
13.7.
.+Numbers of Sequences
13.7.1. For every finite sequence x by card (2).
=
(6,) the cardinality of (n:l,, =+= 01 is denoted
13.7.2. A map s which assigns to every sequence x I a unique sequence (s,(z)) is called an s-function if the following conditions are satisfied:
--
(NS,) llxll = sl(x) 2 s2(x)2 * 2 0. (NS,) s,(x y) 5 s , ( T ) $- j/y//for x. y E I. (NSd s , ( ~ )5 sn(xlIIyIi for 5, Y E I. (NS,) If x E f and card (5) < n, then s,(x) = 0. (NS,) Let M be any set of natural numbers. Then card (31)2 n implies s,(eM)= 1.
+
We call sn(x) the n-th s-number of the sequence x. 13.7.3. For every bounded sequence x the n-th approaimcltion tiurtder is defined by a,(%):= inf (/!z- I / / : I E
13.7.4. Obviously we havc the ‘I’ht.oreni. The mop
a:z -+ (a&)) is an s-function.
i
and card ( I )
0. Let y := ( 7 1 ~ )with tii := if i M and q e := 0 if i M . Then eLll= q.It follows from
+ +
(;’
that card ( M ) < n. If I := zeM,we obtain card ( I )
< n and
+
Ib - 1115 sn(r) E Consequently o n ( ~5) sn(x).This proves that sn(z)= a&). an(x) 5
13.7.6. We now describe an easy method how to compute the s-numbers of a given sequence. Theorem. Let x = (fn) E co. Then there are a one-to-one map n from .Ninto itself as well ns sequences s = (a,) and t = (z,) such that (rl 2 crg 2 2 0, i ~=~jtZll = ... = 1, and z = J,(st). MoTeover, ~ ~ (=2 on ) for 72 = 1,2,...
---
R e m a r k . According to the preceding characterization the sequence (s,(x)) is called the non-ixcreasing rearrangement of x.
13.7.7. Finally, we mention without proof the following P r o p o s i t i o n . Let z = (En)E I. Then n
2’ g ( x ) = sup 13.8.
Iti/:card ( M )
1.
n
.4f
1
Domination Property
13.8.1. A norined sequence ideal [a, a] has the domiitation property if, given and y E a , then n
B
1
1
1s&) 5 z . s k ( y )
for
?L =
2:
E1
1, 2, ...
implies z E Q and a(s) 2 a(y).
13.8.2. First we prove that a, weaker. property holds for every normed sequence ideal. Lemma. Let x , y E f and n
11
sk(x) 5 2’ sk(y) for 1
Then a(z) 5 a(y).
1
7~
= 1 , 2,
...
186
Part 3. Theory of Seyuenre Ideals
Proof. Write x = (&,) arid y = (17%). Without loss of gmerality wc may suppose that >, t22 .-.2 0 a n d q1 2 qz 2 ... 2 0.
c1
We now choose no such that t, = 0 and ?in = 0 for n > no. Let n range over all permutations with n(n) = n for n > no. Furthermore, let t = ( t t L be) a n arbitrnry sequence such that 5, = i.1 for n, 5 no and tn= 0 for n > no. Then the convex hull C of all sequences tQ,y is compact. Suppose that x 6 C. Using a separation theorem we can find a finite sequence a = (a,) such that m
m
2 antn> 1
and
1
2 anyla5 1
for all (7,)6 C .
1
Choose ~t and t such that s,(n) = z,a,(,). Then m
n
m
s (tn-
1 < 2 a,,tn = 1
tn+d
2 mi
n=l
i=l
m
la
f 2 ( t n - En+,) 2 si(a) n=l
2=1
=
tnsn(a) ==
Z (sn(a) - s n + l ( a ) ) 2'
n=l
1
(%(a) - s,+&))
n=l
-
t i
2=1
n
cu
2 2' -
n
m
m
&YV8
i=l
m
m
X sn(a) T n == 2
7naz(n)p17,
1
1
which is a contradiction. Hence x
5 1,
E C . So there exists a convex combination
m
x ==
&tkQxJ1
Since Al,
..., I.,
m
2 0 and
?Lk
=
1, it follows that
1
m
a(%)5
&a(&,$)
= a(Y).
1
113.8.3. P r o p o s i t i o n . Every minimal normed sequence ideal hns the donLinntion l?roperty. Proof. Let [u. a] be minimal. Suppose that n
2 s&) 1
2
E I, y E a, and
n
5 2 sk(y) for n = 1,2, ... 1
Siiice a is ininimal, we have y E co. Moreover,
implies x E c0. Hence, without loss of generality, we can assume that
El 2 & 2 ... 2 0 and vl 2 vz 2 --.2 0.
13. Sequence Ideals on the Scalar Field
Then, by 13.8.2, we get a(Pmx) a(Pmy)for m = 1, 2, ... Given with a(y - Pm0g)5 E and m m.+l
E
187
> 0, there is ?no
m
tk5 2' qk + E
for m
> mo.
mo+l
Ot.herwise, we could find m,
< ?nl
0, there is m, such that n
?t
2 sk(y) 5 (1 + E ) C sk(Pm,y) for n = 1, ...,no. 1
1
Using n
n
n
2 sk(Pn,X) 4 1
5 2'
1
gk(~)
1
we obtain n
sk(Pn*z) 2 (1 1
+
11
E)
sk(Pm,y) for n = 1, ..., no.
1
n
2 sIl(Pn,x)is constant whenever n > no, the above 1 = 1, 2, ... Consequcntly, it follows from 13.8.2 that.
Since
Finally, by 13.5.4, we get z
inequality holds for all
a and a(%)= sup a(Pnz)5 a(y). n
13.8.5. A normed sequence ideal without domination property has been constructed by G. I. Russu [l], [2].
188
Part 3. Theory of Sequence ldeals
13.9.
Lorentz Sequence Ideals
13.9.1. Let 0 < p < M and 0 < q < 00. Then the Lorentz sequence ideal I(,,, consists of all sequences L $ I such t hat
:{
a3
f ( , , q , ( := z)
Cn the case where 0 < p l(p,&)
= p2 2 -.-2 0, C pn = 0 0 , OQ
1=
1
and lim yta= 0. R
and q = 09 an
13. Sequence Ideals on t,he Scalar Field
189
Then the Surgent sequence ideul 3, consists of all sequences x E 1 with
R e m a r k . If @ = ( n - l ' p ) with 1 < p
13.P0.2. T h e o r e m . [&,,s,]
< 00,
then 5, and f, coincide.
i s a trtaximul normed seque?~ceideal which fails to be
,,li,tirnZ.
n
R e m a r k . A sequence r E 1 belongs to
I:
?$" if and only if lini 2' sk(x) 71
I
rpk = 0.
13.10.3. P r o p o s i t i o n . T h e (idjoint normed spquence iden1 [G:, s:] em-ists of all sequences x 5 I such thut m
&5)
y&c)
:=
< 00.
1
13.11. Orlicz Sequence Ideals 13.11.1. Let @ be aiiincreasirigconvex continuous function on 9+ with @(O) = 0 and @(1) = 1. Then the Orlicz sequenee idenl 1, consists of all sequences x = (En) such that m
2' @(ltn//~) 5 1 for some G > 0 . 1
we put
Remark. If @ ( f ) := < p and 1 2 p
13.11.2. Theorem. [I,, l,] 55
i.7
< cu,then 1,
and I, coincide.
a maximal normed sequence idenl.
R e m a r k . If @ satisfics the so-called (&)-condition, then [l,, lo] is minimal, as ell.
13.11.3. Finally, we nieiition that 1; is also an Orlicz sequence ideal.
13.12.
Notes
scguence ideals, that means normal permutation-invariant linear spaces of bounded sequences, \ \ w e investigated by many authors. The most results of this chapter belong t o the folklore of functional analysis, and we are riot able t,o give exact references. The int,erest,ed reader should consult D. J. H. GARLING[l:].The concept of a minimal and maximal normed sequence i!lcitl can be found in B. S. XITJAWX [2]. The domination property --as introduced by P. A. V-ILDERON [l]; see also B. S. MITJAGIN [3]. Further information about Lorentz sequence i(1eals are given in [BUT, pp. 181-1871 and [TR.I, pp. 131-1351. For the theory of Sargent sequence ideals we refer t o [GOH, pp. 139-1501. and Orlicz sequence ideals are treated i n LLIN, p. 561. The standard reference for Orlicz function spaces is the monograph [KRA]. ~~ecominendations for further reading-:
J. I. GRIBANOVI l l , G. KOTHE/O.TOEPLITZ 111, G . I. Itc-ssu [ll, 121, W. L. C. SARGENT [l].
14.
Operator Ideals and Sequence Ideals
I n this chapter we deal with operator ideals generatt-d by a n additive s-function and a scalar sequence ideal. The class Gf) consists of all operators S such that (Sn(S))E a. For special s-functions the above definition goes back to I. A. NOVOSELSILIJ and the author (1963). The main interest is devoted t o those operator ideals (5); which correspond to the classical ideals I,. We state their basic properties and investigate the relationships between operator ideals GF) defined by approximation numbers, Gelfand numbers etc. Moreover, the same method is used for generating operator ideals by the entropy numbers. Conversely, for every operator ideal 8 and fixed exponents u and v we may consider the ideal of all scalar sequences such that the corresponding diagonal operator 5). This leads to the interesting concept of a limit order. belongs to We also define so-called small operator ideals and give the most important examples.
14.1.
GP’-Operators
14.1.1. Let s be an additive s-function and let Q be a sequence ideal on the scalar field. An operator S E B(E,F ) is called an G,bs)-operatorif (an(&)) E a. The class of these operators is denoted by St’.
14.1.2. Theorem. Gf) i s an operator ideal.
+
Proof. Let S,, S, E G f ) ( E ,F ) and put S := S , S,. Then it follows from 13.1.3 that, (s,(SL), s,(Si),s,(S,),s2(Si),...)E a for i = 1,2. Cy S , ~ - ~ (5S sn(SI) ) sn(S,) we hare (‘I(’)?
sl(s)> ‘3(’),
s3(S)7
’**)
+
E a.
Hence (sdf% s,(S),s3(S),%(S), * * .)
This proves that XI evident.
E a*
+ S, E GF). So (01,)is satisfied. The remaining properties are
14.1.3. P r o p o s i t i o n . For every (compbtely) symmetiic additice s-function €he operator ideal Gf) i s (completely) symmetric, as well. 12.1.4. P r o p o s i t i o n . Por every regular additive s-ftinction the operator ideal G!’ regular, as well.
14.1.5. P r o p o s i t i o n I . For every injective additive s-fwnction the operator ideal 6;’ i s injective, as well.
14. Operator Ideals and Sequence Ideals
191
P r o p o s i t i o n 2. For every surjective additive s-function the operator ideal G t ) is surjective, as well.
14.1.6. Let [a, a ] be a quasi-nornied sequence ideal on thc scalar field. Then we put
Sf’(S) := a(s,(S)) for S 6 Gf). 14.1.7. Using the method of 14.1.2 it can easily be seen that S t ) is a quasi-norm. However, we do not know whether G:’ is also complete. P r o b l e m . Is [G:), SF’] a quasi-normed operator ideal?
14.1.8. We now give an affirmative answer to the preceding question in a rather general case. T h e o r e m . Let [a, a] be a maximal sequence ideal on the scalar field. Then [Gr’, SF)] i s a qwi-normed operaior ideal. Proof. By 13.2.8 we may suppose that a is a p-norm. Therefore the criterion 13.5.4 can be used. Let (8,)be a n SF’-Cauchy sequence of operators 8, E G:’(E, F). Then there exists 8 E e ( E ,B’) with 8 = ~ ~ . ~ ~8,. - l iIt m follows from k
\$‘e now obtain ( ~ ~ -( Sk)) 8 E n and a(s,(S - 8,)) 5 & for k 2 ka. Therefore S 6 Gf’(E,F ) and X = S&s)-li~:i S,. So the completeness of [Gf’, St’] is proved. k
14.1.9. The next result is an improvement of the preceding theorem. P r o p o s i t i o n . Let s be a maximal additive s-function and let [a, a] be a maximal quasi-normed sequence deal on the scalar field. Then the quasi-normed operator ideal [Gf), St’] i s maximal, as well. Proof. Suppose that S E 2(E,P)belongs to the maximal hull of Gf).Let (tn)E c0 with Itn/5 1. Theii, given E > 0, by 11.10.3 we can find S E 6(Eo,E ) and B E % ( F ,Fa) such that ( ( X /5 ( 1, ((B(( 2 1, and
192
Part 3. Theory of Sequence Ideals
Now it follows from (s,(RXX)) < a that (rnsn(S))E a even for all ( ~ ~ (E8a )and ) thcrefore S’ E G f ) ( E ,F).Moreover, we have
(T,)
E co. Iiencr
S 3 S ) = a(sn(S))= sup { a( ( w, ( S) ) ) :(z,) E co, I~,I 5 11
5 (I 5 (I where
+ E) sup {a(s,,(BSX)): //XI15 I, /IS// 2 1) $- s) sup
{S&S’(BSX): IIXIl I 1, 1 1 ~ 1 15 11,
X @(€to, E ) and R E @(F,Fo).This completes the proof.
14.1.10. P r o p o s i t i o n . Let s be a n ultrastable ndditive s-function and let [a, a] be n ,tiaximal quasi-normed sequence ideal on the scalar field. Furthermore suppose that a i s weakly lower semi-continuous. Then the quasi-normed operator ideal [Gt’, Sf’] i s ultrastable, as well. Proof. We consider a n SF)-bounded family of operators 8, E GF’(Ej)F b ) . Put where & := lini sk(S,).Then
.c = (&))
U
Using 13.2.8 and 13.6.4 we obtain z (8,)n E G ~ ) ( ( E ~(F,)tl). ) ~ , Moreover,
St)((S,),)
=I
c a. Now
it follows froin S ~ ( ( S ,5 ) ~&) that
a(SL((S,)U)) 2 liin a(sk(S,))= lim S!$(S,).
u
11
14.1.11. Finally, we deal with thc operator ideal @(’) generated by ail additive s-function s and the scalar sequence ideal c0.
14.1.12. Theorem. TIP qerator zdeul
is closed.
Proof. Suppose that 9 t Z(E,3’)belongs to the closure of W). Then there exists E0 f @(*)(E, F ) with I/S - Sol/5 E . We now choose a natural number no such that s,(S,) E for n 2 no. Clearly, s,(S) 5 sn(So) l/S - Sr,Il 5 2s for n 2 no. This proves that S E @(”(E,F ) .
+
14.1.13. P r o p o s i t i o n .
==
6. WC) =R, and W d )= R.
Proof. The first equation is evident. The remaining ones follow froin @ma= R and @sup = 8 as well as from the fact that @ ( c ) and @(d) is the injective hull and the surjective hull of @fa), respectively. R e m a r k . A direct proof of the last equation can be found in [PIE, p. 1461.
14.2.
%,-Operators and @,-Operators
14.2.1. Let s be a n addit,ive s-function and 0 < p is called a n Gt)-operator if (s,(S))E I,. We put
The class of these operators is denoted by G!).
< m.
An operator S < Z ( E , PI’)
14. Operator Ideals and Sequence Ideals
14.2.2. Theorem. [GF), SF)] i s
LC
193
quasi-normed operutor idenl.
Proof. The assertion follows immediately from 14.1.8. However, we are interested in estimating the constant in the quasi-triangle inequality. For this purpose let S,, S2 E G:){E, F ) . Put x := Zl’P niax (S’F1,1). Then
s X[St’(S,) + s;’(s*)]. Remark. We see from 6.2.5 and the preceding estimate that [GF), St’] is q-normed for l / q := 111, + 1 and 1 5 p < 00. This result, cannot be improved; cf. 18.5.1 and 18.6.1 (remark).
14.2.3. Theorem. Suppose that s is n multiplicative and additive s-function. If < p , q < 00 and 11‘ = l / p l / q , then GE’ o Gf’ E GP).
+
0
Proof. For S
c GF)(F,G ) and T E G:’(E, F ) we have
This proves that ST E GF’(E,G). 12.2.4. Let us agree, for simplicity, that [Up, A,] denotes the quasi-normed ideal of GF’-operators generated by the approximation numbers. Operators belonging to this ideal are frequently called Up-operators.
14.2.5. Theorem. The quasi-normed operator ideal [Up,A p ] is completely symmetric.
Proof. The syinrnetry follows from a,(S‘) 5 u,(S); cf. 11.7.3. We now suppose that S E aF“‘(E,F).Then 8‘ E B(F’,E’) and therefore S E a(E,P).So: by 11.7.4, we get a,(&’) = w,(S). This proves that By]= [Up,Ap1. 14.2.6. As a consequence of 11.10.9 and 14.1.10 we have the Proposition. The quasi-nornted operator ideal [a,, Ap] i~ ultrcistable.
12.2.7. Theorem. The quasi-normed operator ideul
[Up. A,] is muxirnul.
Proof. We know from 14.2.5 that [a,, Ap] is regular. So, the maxiniality is implied by 8.8.7.
14.2.8. Lemnia 1. For every S E Up@, E’) and that Ap(S - L ) r F . 1s Yietsch, Operalor
E
>0
there existx L E
a@,F ) such
Part 3. Theorv of Seguelice Ideals
194
Proof. Choose m with
Then 2m
mazm(S)P 5 2’ a k [ S ’ ) p
0: we choose an operator L, g ( E , F ) with rank (L,) < n and jlX - L,J 5 (1 + E ) n,,(iS)for n = 1, ..., r , where r := rank (8). Put
N :=: N ( 5 ) n S ( L , ) n and
+-
M := 111(5) N ( L , )
.- n W(L,)
+ - - - + M(L,).
Then N E Cod ( E ) and iM E Dim ( B ) . Obviously there exist unique operators Xo, Llo, .,L,, E g ( E / N ,M ) such that the following diagram commutes:
..
Since the operator norin is injective and surjective, we have
C o n j e c t u r e . The quasi-nonned operat,or ideal
[up,A,]
is minimal.
1%
14. Operator Ideals and Sequence Ideals
R e m a r k . Since S E U,(E, P) can be represented in the form S = AJim 8, n
with S, E g ( E , P), it follows from A?'"(S, - 8,) = A,@, - S,) and the com= SC,. But we do pleteness of a,"i"(E,H) that S E n,"l"(E,P). Consequently not know whether the quasi-norms and A, coincide on the whole ideal. This lack is %I consequence of the quasi-norm catastrophe; cf. 6.1.9.
%rin
14.2.9. The next result is implied by the definition of Gelfand numbers and Kolmogorov numbers, respectively. Theorem.
[a?, Ainj] P = [Gt), SF)]
and
[Uy,A y ] = [Gkd),Sr)].
14.2.10. By 11.7.7 we have the Theorem. The quasi-norm& operatw ideals [@$, AFj] and to each other.
[WF, A r ] are dual
14.2.11. P r o p o s i t i o n . Suppose that P is a n internwdiaie space of {Po,F,] posse-ssing J-type 8 . Let 0 < po,p , < CQ and l i p := (1 - 8)/po+ e / p l . I f S E E(E, HA),then S 6 %Ej(E, Po) and S E UkJ(E,F,) imply S E U:j(E, 8). Moreover,
AFj(S: E + F ) 5 2llpA;j(S: E -+Fo)l-o A z j ( 8 : E +-F,)*. Proof. By 11.5.S we have
14.2.12. Using tho same method from 11.6.8 we get the P r o p o s i t i o n . Suppose tlzat E i s an intermediate space of {Eo,El} possesSing K-type 0. Let 0 < po,p 1 < cy) and l / p := (1 - 8)/po 8/pl. I f S E B(E,, F),then s' E %Er(Eo, F ) and S E %:,'r(El, F ) imnply S E U F ( E ,F ) . Aloreover,
+
AF(S:E
3
F) 5 2llpA3S: E ,
3 F)l-O
Ar(S:El+ F ) e .
14.2.13. TOprove the following theorem we need an eleiiieritary 01
Leiiinia. Let 0 < p < p o < w and l / q = I l p - lipo. If 2 0, then (nl/poon)E 1., 2 o2 2 Proof. Using
13*
(ol$)E
1, such that
196
Part 3. Theory of Sequence Ideals
we get
As a consequence of 11.12.2we now have the T h e o r e m . Let 0 < p
< 2 and
l / q = 1/p - 1/2. !"en
5 a,, and
S C P 5 2Iq.
R e m a r k . It will follow from 14.4.9 and 14.4.10 that, given p , the exponent q cannot be improved.
14.2.14. Lemma. Let 0 < po < w. If (en)E I,, ( n l k o ,E) G.
and el 1 c2 2
--.2 0,
then
Proof. Choose m such that
Then, for n 2 2m, we get n
nc?
5 2(n - m ) crp 2 2 z afG"5 B e p ~ . m+l
This proves that lim n%sn = 0. I
As a supplement of the preceding result we now have the Theorem.
ulp G 0 and U y 5 0.
R e m a r k . Enflo's counterexample yields OinJ$ 8 and GSur$ 0 . Using the same construction it is possible to show that even $ (5 and $ 0 for 2 < p < m; cf. K. D. E ~ S T E [l]. N
up
14.2.15. Let
11s
agree, for simplicity, that
[Bp,Hp] denotes the quasi-normed
ideal
of Gr)-operators generated by the Hilbert numbers. Operators belonging to this
ideal are frequently called &wperatms.
14.2.16. As a consequence of 11.7.8and 14.1.3 we have the T h e o r e m . The quasi-normed operator ideul
[ap,€Ip] is completely symmetric.
14.2.17. The next result follows from 11.10.13and 14.1.9. Theorem. The quasi-ncurmed operator ideal
[a,, €IP] is maximd.
14.2.18. A proof of the following result is given in [GOH, p. 371 and [HAR, p. 89.1 Lemma. Let & 2 E2 2 ?a
n
1
1
--.and q1 2 yl2 >= -.such that
x l k s z ? ?f ko r n = l . 2 , ... If @ is a conwx function on the real line with lim @({) = 0, then .Lt-
A
n
X@&) 5 .Z @h) for 1 1
n = I,%
...
m
11. Operator Ideals and Sequence Ideals
197
We are now prepared t,o check the Theorem. Let 0 < p
1
Lot
@(E)
l / q = l/p - 1. Then
&:, E U$ and Sj, 5 UF.
&,(It, P ) . We know from 11.12.3 that
Proof. Let S n
< 1 and 98
log [k-lck(S)] 5 2 log [ e 7 ~ ~ ( 8 ) ]for n = 1, 2, ... 1
:= exp
( p t ) .Then it follows from the preceding lemma that
ra
n
1
1
3 [ k l c k ( S ) ] p5 e p 2 hk(S)p
for n
= 1,
2,
...
Using 14.2.13 (lemma) we now get (ck(S)) I,. By 14.2.9this means that S E U t j ( E ,3‘). The second inclusion can be checked in the same way. Hence Sj,
%rj.
14.3.19. C o n j e c t u r e . Let, 0 < p
< 1 and l/q = l/p - 1. Then Sj, & U,.
14.2.20. Finally, we mention that the more general quasi-norined operator ideals generated by the quasi-nornied Lorentz sequence ideals [l(p,9)7 l,,,,] play an important role in the theory of Sobolev’s embedding maps. We mention in particular the case where q = 0 0 ; cf. [TRI].
[6&,, S&]
14.3.
(E,-Operators
14.3.1. An operator S
B(E,F ) is called an @:,-operatorif
(en(&))E 1 , with 0 < p
< m.
We put
Here the norming constant
E,
is choosen such t,hat Ep(Zcf) = 1.
The class of these operators is denoted by @ ., 14.3.2. Analogously to 14.2.2 we have the Theorem. @ [,
E,] is a quasi-nmed operator ideal.
R e m a r k . In the above definition we can also use the inner entropy numbers. Then we get the same operator ideal, and
yields a n equivalent quasi-norm which is sometimes more convenient to work with ; cf. 14.3.4 and 14.3.5. Here the norming constant rp:, is choosen such that Fp(Zx) = 1. 14.3.3. The next result follows froiu 12.1.6. T h e o r e m . If 0
< p , q < cx) and l J r = l/p + l/q,
then (E, o (E,
14.3.4. We now state an immediate consequence of 12.1.9. Theorem. The operator ideal @,
is
wmximd.
(Er.
198
Part 3. Theory of Sequence Ideals
14.3.6. Applying 12.1.8 we get the Theorem. The operator ideal Cr?, is injective and surjectiw.
14.3.6. Until now no relationship between e,(S) and e,(S) semis to be known. P r o b l e m . Is the operator ideal Gp completely symmetric?
14.3.7. For completeness we mention the following interpolation property which follows from 12.1.11. P r o p o s i t i o n . Suppose that E is an. intermediate space of {Eo,E l ] possessing K-type 8. Let 0 < p Q , p l< 00 and l/p := (1 - 8)/pQ O/pl. If S E E(E,, P ) , then S E (.fpo(Eo, P)and S E @p,(E1,P)imply S E GP(E,F ) .
+
12.3.8. Using 12.1.12 we analogously get P r o p o s i t i o n . Suppose that F i s an intermedinte space of {PQ,PI) pamessing J-type 8. Let 0 .< po, p1 < 00 and l/p := (1 - 8)/po 8/pl. If S E E(E,F A ) , then S E (.fP,(E, Fo)and S E (.fp,(E,F1)imply S E e P ( E ,F ) .
+
14.3.9. Obviously 12.3.1 yields the T h e o r e m . Cip 5 $jpfor 0
< p < 00.
14.3.10. The following inclusions can be derived from 14.8.15 and 14.3.9. See also 12.3.2. T h e o r e m . Let 0 < p
< 1 and
l/q = l/p - 1. Then
14.3.11. Finally, we state a converse inclusion. Theorem. Let 0
< p < p < m. Then aP& eq.
Proof. Suppose that S E Mp(E,F). Then there exists a positive constant c with
arn(S)d For every n
C
mllP
for
ni. =
1, 2, ...
> 1 we determine m,by
n-1
plog?l.
<msp-
n-1
log n
+ 1.
It now follows from 12.3.3that
Hnece (.tg(S))E Iq and therefore S E G,,(E, P).
14. Operator Ideals and Sequence Ideals
Remark. Since Gqis injective and surjectivc, we also have UF
14.3.12. Conjecture. Up 5 Gpfor 0 < p 14.4.
199
B, and U r 5 Bq.
< 00.
Limit Order
14.4.1. For every operator ideal 9l the limit order A(%, u, v) is defined to be the and 1 2 u,v jbo. infimum of all ii2 0 such that Dl E U(Z,, l"), where O&) :=(r2tn) 14.4.2. The importance of the above concept comes from the following property. Proposition. Let % be an operator ideal. Then A(%, u,v) is the infimum of all l / r 2 0 such that every diagonal operator of the form AS&) = (on&,)with (on)E I, helongs to Q). Proof. Suppose that r has the above property. If 3. therefore D1 E %(la, 5). Hence l/r 2 A(%, u, a). G~
Conversely, let (on) I, with l/r > 2 o, 2 2 0. Then it follows from
---
A(a,u, v).
> llr,
then ( r 2 E 1,) and
Clearly we may assume that
that on 5 an-llr. So D,,, E ?ll(l,,1,) implies S E N(lu, l,,).
14.4.3. The next characterization is very useful in computing special limit orders. Theorem. Let [a,A] be a qwcsi-normed operator ideal. Then infimum of all ii 1 0 such that
A ( I : 1: --f 1:) en1 for n = 1,d, where e 2 0 is some constant.
?,(a, u, v) is
...,
the (* 1
Proof. According to 6.2.5 it is enough to check tho assertion for every p-normed operztor ideal. First let us suppose that (*) holds. Put
Nk:= ( n : 2k 2 n c 2k+l}, and let Qk be the diagonal operator generated by the characteristic sequence of X k . Then A ( Q k : 1, -+ 1,) 5 e [card (Nk)ll= ~ 2 ~ ~ . For E
> O we get,
2' A(2-(14-&)kQk)p 5 Q P x 2-8pk < m
03
0
0
00.
Consequently 03
:=
2-(i+dkQk 0
Telongs to %(la, 5). Since n-(l+&) 5 W J + E for ) k n E N,, we have DJ+* E 1) This means that i. E 2 I.(%, u , v).
+
5).
200
Part 3 . Theory of Sequence Ideals
Conversely, let A. > I.(%, u, v) and put r := l/l.. Then the inap 13 assigning to every sequence (an)E I, the diagonal operator S E 91(Z8, 1,) defined by S(&J := (a&) is closed. So there exists a constant e 2 0 such that for all (a,,) E 1,.
A(&: I , -> 5) 5 &(a,) In particular, if
...,En, t&, ...)
P&,
:= (51,
... l a , 0, ...) )
>
then we have
A(]: 1; -+ Z;!)
=
d(P,:I,
--f
1,)
5 en1'' for
n = 1 , 2. . ..
This proves (*).
14.4.4. We now consider a first example. Lein ma.
Hence we have the Proposition.
14.4.5. In what follows the behaviour of A(%, u, v) is graphically represented by inearis of diagrams in the unit square. We shall use the coordinates l/u (horizontal) and l / v (vertical). I n the left-hand diagram we plot the level curves, while the algebraic expressions of I(%, u;v) are indicated on the right.
For the ideal of all operators we have the diagram:
14.4.6. The next result is trivial. Proposition. Let
A(%
0
and ?8be operator ideals. Then
d,u , 10) 5 I,(%,
21:
v)
+ I.(%,
c, 2 4 .
Remark. In particular, we have
q2€,21, v ) I q2,u , uo) + I.(%,
uo, %lo)
+ 42,
2'0, 2')
This estimate is the basic tool for all computations which will follow later on.
14. Operator Ideals and Sequence Ideals
201
14.4.7. Proposition. Let 2I be an operator ideal. Then I.(%mi", u, v) = I.(%-,
u, v) = ).(a,u, v).
Moreover,
A(%-',
u,w) = A(%, w', u').
14.4.8. Let us mention that the limit order is a continuous function of 1/u and 1j.l.. More precisely, we have the
Lemma. [A(%, u, v) - A(%, uo, vo)l S jl/u - l/uol
+ Il/v - 1/d.
Proof. The inequality follows immediately from 14.4.4 and 14.4.6 (remark). 14.4.9. Proposition.
for l d u 1 2 ~ v I c o .
5 IL 5 cc,then by 11.11.5 we have
Proof. If 1 5 ak(I:
):Z
3
= (12
-- k -+ 1)l'r for k
= 1,
...,
72:
where l/r := l/v - l/u. Consequently
Here x means that both sidea are of the same order; cf. index of symbols. Therefor
A(%,,, u, v) = 111, - liu We know from 11.11.8 that
+ l/v
for 1 5 v
u S
00.
202
apm
Part 3. Theory of Sequence Ideals
The above result is represented in the following diagram :
1 -
2
P
1
Remark. It has been recently proved by B. CmL/A. PIETSCH [2] that in t,he case where 0 < p < 1 and 1 5 u 2 2 2 v 5 00 we have
A(%p, u, v) = max (1/p - l/u
+ 1/2, 1lp - 1/2 + l/v) .
The same result should be true even for 0 < p
A(%p, u, v) = max (l/p - 1/u whenever 0 < p
'
+ l / v 5 Acep,u, v).
On the other hand, we have seen in the proof of 12.2.2 that e k ( l : +-Zk) 5 8
log (n
k
-+ 1)
for k = 1,
...,n .
=.
(4)
u)4
Part 3. Theory of Sequence Ideals
Xoreover, en+*(I:1% +-1:) = 112. Using 12.1.5 we get
s cnl’P-1 log (n + 1) , where c is some positive constant. This yields A(@,, 1, 00) 2 1/p - 1. from 1, =) - A(2,m, v) A(@,, u, v) 5 A(2,u,1) that A&, u, v) 2 1/p - 1/u l/w.
+ +
it f ~ h w s
(2)
The above result is represented in the following diagram :
14.4.13. Analogously we get the
Proposition. Let 1 5 p
< m. Then
u,v) = max (1/p - l/u
+ l/v, 0 ) .
This yields the diagram :
14.4.14. The limit order of the operator ideal 4, with 0 < p < 1 has been computed by B. CARL/Q.BETSCH [2] recently.
14.5.
Small Operator Ideals
14.5.1. An operator ideal is finite dimensional.
is called mudl if %(E,F ) = B(E, F ) implies that 1or F
14.5.2. Proposition. Let s be an injective,surjective, and additive s-fuwcthz. T?wk Gg) .is ,mall.
14. Operator Ideals and Sequence Ideals
205
Proof. If G,a")(E, B') = B(E, F) then there exists a constant e 2 0 such that S:)(S) 5 e llSll for all S E O(E, F).Assume that E and P are infinite dimensional. By E.2.2, for every natural number n, we can find A, E O(E,lg) and Y , E O(%, F) with 1 5 q(A,) 5 /[An[[ 5 2 and 1 5 j ( Y , ) 5 l[Ynll 5 2. Applying B.3.3 and B.3.7 we have the obvious diagram
where 1IA;Jl
5 1 and 1lY;~Il (= 1. Put AS, := Y,I,A,. Then
sk("9R)
= S d J n YnolnAnoQn) = 8 d Y n o l n A o )
I IIYTJI 8dYnJnAnd IIAGVI I din) == 1 for k = 1,
...,n. This yields nl'v 5 S!)(&)
2 e [IS,,ll2 4p , which is a contradiction.
14.5.3. The following main result is an immediate consequence of 11.2.3, 11.5.5. and 11.6.5 as well as of 11.7.9. Theorem. The operator ideals Up,4[2, and
%r are small.
14.5.4. Finally, we state a problem which goes back to A. GROTHENDIECK.
Conjecture. The operator ideal !It is small. Remark. Some partial results are due to W. J. DAVIS/W. 3. JOHNSON [I] and I. I. TSEITLIN El].
14.6.
Notes
J. W. CALKM observed that operators on the separable Hilbert space belonging to a given ideal can be characterized by the asymptotic behaviour of their a-numbers. Since the concept of 8-numbers has been extended t o operators between Btlnach spaces, the same construction yields operator ideals on the class of all Banach spaces. Starting from the Kolmogorov numbers and the approximation numbers I. A. NOVOSELSEIJ [I] and A. PIETSCH [2] have defined special kinds of Gf)-operators. See also W. B s n H s s D T [l]and B. CARL/A.PIETSCH [I]. Interpolation properties of Bas)-operators are extensively treated in [TRI, pp, 107-1171. The [14]. Furthermore, A. PIITSC'H important notion of a limit order first appeared in A. PIETSCH [19] proved that all operator ideals Up with 0 < p < 00 are small. Recommendations for further reading:
CH. CONSTANTIN [ 2 ] , W. J. DAVIY/W. B. JOHNSON [l], R. A. GOLDSTEIN/R. SAEKS[1], C. V. HUTTON [Z],H. KOXIQ[l], A. PIETSCH [18], [20], B. ROSENBERGER [3], I. I. TSEITLIN [I].
15.
Operator Ideals on Hilbert Spaces
The theory of ideals in t-he operator algebra of the separable infinite dimensional Hilbert space was created by J. W. CALEIN in 1941. Excellent preseatations are G. KREJNand R. SCHATTEN. This given in the monographs of I. C. GOHBERG~M. chapter deals with operator ideals on the class of all Hilhert spaces. First we show that each proper operdtor ideal is uniquely determined by its component on 1,. Then a one-to-one correspondence between scalar sequence ideals and proper operator ideals is established. Unfortunately, we do not know whether this relationship also liolds in the quasi-nornied case. So we must restrict ourselvcs to nornied ideals haviiig the domination property. The main part of this chapter is devoted to so-called Qp-operatorsfirst defined and R. SCHATTEN about 1946. In the case where p = 1 these by J. VON NEUMANN are the operators of tracc class (nuclear operators), and p = 2 yields the classical ideal of Hilbert-Schmidt operators. Finally, we extend operator ideals from the class of Hilbert spaces to tho class of all Banach spaces. Xost of them have an infinity of different extensions. On the other hand, we will see later on that the ideal of strictly nuclear operators admits a unique extension. All concepts in the theory of operator ideals on Hilbert spaces are analogously defined as in the case of Banach spaces.
15.1.
Operator Ideals
15.1.1. Proposition. Every (quasi-normed) operator ideal on Hilbert spaces 6 conzpletely symmetric. Proof. The asscrtion follows from D.3.1 and the fact that So = 8; for every diagonal operator Soacting in L2(f2,p).
15.1.2. The next result states that every (quasi-nornied) operator ideal on Hilbert spaces is injective; cf. 4.6.9. Proposition. Let U be an operutor ideal. For every injection J E 2(Ko,K ) and every operator So E 2(B,K O )it follows from JSo E U(H, K ) that Xo E % ( H ,K O ) . Proof. Put Qy:= J-lPy, where P is the orthogonal projection from K onto H ( J ) .Then QJ = I,*. Xence So = Q(JSo)E % ( H , K O ) .
15.1.3. The surjectivity can be checked in the same way. Proposition. Let 3 be an operator ideal. For every surjection Q E i?(II,H o ) and every operator 8, E 2(Ho,K ) from SoQE %(H, R)it follows that So E %(I& K ) .
15.1.4. In the theory of operator ideals on Hilbert spaces there are further simplifications due to the fact that the metric approximation property holds; cf. 10.3.
15. Oyeutor ldeals 011 Hilbcrt Spaces
15.2.
205
Proper Operator Ideals
13.2.1. We start this section with a useful characterization. Theorem. An operator S E B(H, K) isapproximableijalzd onlyif ( ( S X e , , Ye,,))c co for all operators X E B(12, H ) and Y E B(l2,K ) . Proof. Suppose that the above condition is satisfied. Given E > 0, we choose maximal orthonormal families ( x i ) and ( y i )with l(Szi,yi)l 2 E for all i E 7. Observe that I is finite. Otherwise we take il,i2,.. . I . Let m
I ( &:= ) 2 taxi,, and
m
Y(7,‘):= 2 vnyi,.
1
1
Then it follows that lim (Sxi,, yi,) = lim (SXe,, Ye,) = 0 , n
n
which is false. Put
A := 2 xt @ xi and B :=2’ y: @ yi, I
I
l i ( I K - B ) S ( I H- A ) ] ]> E . Then thereare x E H and y f K with [((IK- E ) S(IH- A ) x , y)I > E 11x11 11y11. Put xo := x - A z and yo := y - By. Obviously, we may suppose that ll.rOl]= 1 and llyoll = 1. It follows from 111, - All = 1 and llIK - Bl/ = 1 that (/zjj2 1 and Jjy/l2 1. Hence /(Sxo,yo)) > E. So we e m
Let us assume that
enlarge ( x i ) and (yi)b y adding zo and yo, respectively. This contradiction implies
11s - S A - BS + BSAll = /1(19- B ) S(IH - A)ll 5 E . Hence S E @ ( H ,K ) . Conversely, let S E 6 ( H , K ) , X E 2(Z2,H ) , and Y f f?(Z2, K ) .Then Y*SX E a(&, &) yields Y*SX E B(Z2, Z2). Since (en)is a weak zero sequence, we get lim jl Y*SXe,,jl = 0. n
Using I(SXe,, Ye,)l 5 IIY*SXe,,II it followsthat lim (SXe,, Y e , ) = 0. n
15.2.2. The next r e s d t is of great importance. Theorem. Every proper operator ideal on Hilbert spnce.s colzsists of approxirmble operaiors only. Proof 1. Let 2l be a proper operator ideal. Then it follows from %(Z2, 7,) =/= f?(Z,, Z2) and 5.2.2 that 9€(Z2, 1,) G 6(12, Z2). Consequently, for S E %(H, K ) , X E e(4,H ) , and y E B(12, K ) we have Y*SX @(&, Z2). Finally, 15.2.1 implies liin (Y*SXc,, en) = 0 and therefore S E @(H,K ) . So % ($5. n P r o o f 3. Let U be a proper operator ideal. Then SOW 5.4.6 implies 8c - ~ U C l O S= $p= gc10s = @ *
2lclm
is also proper by 4.2.8.
15.2.3. P r o p o s i t i o n . Let U be cc. proper operator ideal on Hilbert spaces. Then S E 2 ( H , K ) belongs to 2l if and o?tl!j if Y*SX E a(&,I,) for nll operators X 2(1,,H) ancl Y E f?(12, K ) .
208
Part 4. Theory of Sequence Ideals
Moreover, for every quasi-norrn A defined on 91 we have A(8) = sup (b(Y*SX):!/XI/5 1 and liYI] 5 1).
(*I
Proof. Suppose that S E E(H, K ) satisfies the above condition. Then S is approximable by 15.2.1 and 15.2.2. Consider the Schmidt factorization of S described in D.3.3. Then So = Y*SX E 'u(Z2,I,) implies S = YSoX* E n ( H , K ) . This shows the sufficiency of the given property. Its necessity is trivial. The forniula (*) is an inimediate consequence of D.3.3 and (QOI,).
16.2.4. By the same considerations as in the preceding proof we are also led tQthe Proposition. Let% be aproper operator ideal on Hilbert 8paLe8. Then 8 EE(H, K ) belongs to U if and only if 8 = YS,X* with So E a(&, Z2), X E f!(Z, H ) , and Y E B(h, K). illoreover, for every quasi-norm A we have
A(S) = inf IlYll AWO) IlXllY where the infimum is taken mer all possible fnctorizatims.
16.2.5. As a consequence of 15.2.3 or 15.2.4 we state the fundamental Theorem. Every proper (qua-G-nornzed)operator ideal 012 Hilbert spaces i s uniquely determined by its component on 1,.
16.2.6. Using the characterizations given in 15.2.3 or 15.2.4 as a definition we can extend every proper operator ideal a(12, Z2) to the class of all Hilbert spaces. SO we obtain the following improvement of the preceding result. Theorem. There i s a one-to-one correspondence between the proper (quasi-izornied) operator ideals an the class of all Hilbert spaces and the proper (qua8i-normd) operator ideals on the separable infinite dimensional Hilbert space.
15.3.
Operator Ideals and Sequence Ideals
15.3.1. Let U be an operator ideal on the class of Hilbert spaces. Then 1% denotes the set of all sequences (u,,) E 1 such that the operat,or S E B(Z2, I,) defined by S(t,) := (u&J belongs to 'u. 15.3.2. Obviously we have the Theoreni. 1%i s a sequence ideal.
15.3.3. Let a be a sequence ideal on t,he scalar field. Then e, denotes the class of all operators S E E such that (s,,(S)) E a. Remark. Let us recall that, by 11.3.4, there exists only one s-function on the class of Hilbert spaces which is of course completely symmetric, maximal etc.
15.3.4. The next statement is a special case of 14.1.2. Theorem. B, is an operator ideccl.
15. Operator Ideals on Hilbert Spaces
209
L5.3.5. Summarizing 15.3.2 and 15.3.4 we obtain the main result of this section. Theorem. The rules
?.+la
and
a+f,
define a one-to-one correspondence between all proper operator ideals on the d m s of €filbert spaces and all proper sequence ideals on the scalar field. Proof. The assertion will follow from the fact that S E U(R,K ) if and only if
(4s))E b. To check this conclusion we consider the Schiiiidt factorizations S = YSoX* and So= Y*SX described in D.3.3. Then, by on = s,,(S0)= sn(S), the following statements are equivalent: s E ?.(IT, 10, soE ?.(12, 12), (a,) E IS, (s,(s))E 1%. R e m a r k . Clearly there is a one-to-one correspondence between all operator idcals in the separable infinite dimensional Hilbert spacc and all sequence ideals on the scalar field; cf. 15.2.6.
15.3.6. Let [a,A] be a quasi-normed operator ideal on the class of Hilbert spaces. Then we put la(s) := A(S) for s = (a,) E 1%, where the operator S E 2(Z2,1,) is defiiied by the equation S(:,,) := (antn). 15.3.7. Obviously we have the
Theorem.
[Is, lA]is a quusi-normedsequence ideal.
15.3.8. Let [a, a] be a quasi-nornied sequence ideal on the scalar field. Then we put := a((s,(S))) for S t Ga.
Sd@)
15.3.9. As a special case of 14.1.7 we now forinulate the open P r o b l e m . Is [Go,S,] a quasi-normed operator ideal?
15.4.
Domination Property
15.4.1. A normed operator ideal [a, A] has the domination. p r q e r t y if, given S E O(H, K ) and T E U(H, K ) , then n 1
iiiiplies S E
n
sk(S)5 2 sk(T) for n = 1,2, ... 1
a(H,X )and A(S) 5 A( T) .
15.4.2. Obviously we have the Theorem. Let [a,A] be a normed operator ideal with domination propert!!. Then [la- 1 ~ i ]s a normed sequence ideal possessing the dominution property, a.s well. 00
15.4.3. Lemma. I f S E G1&, Z2), then 1
I(Se,, en)i 5 S,(S).
Proof. Consider the Schmidt representation
1d
Yietecli. Ooerator
210
Part 3. Theory of Sequence Ideals
with c1 1 0, 2 -. 2 0. By 11.3.3 and Bessel’s inequality it follows from 00
(Sek, 9)= 2‘ n=l
a,(ex-, 5,) ( Y n , el,)
that m
W
5 2’ un
I(Sek, k= 1
n=l
w
2 I(%, k=l
5,)
ek)i
m
s z: 0, = S,(S). n=l
We are now able to check the fundamental Theorem. Let [a, a] be a w m d sequence ideal with domnimtbn property. Then an qerator S E B(H, K ) belongs to 6,i f and only i f ((SXe,, Ye,)) E a for all operators x E 2(Z2, H ) and Y E f i ( Z 2 , K ) . I n this m e ,
SE(X) = SUP (a((SXe,, Ye,)): IIXII 5 1, IIYII 5 I}. P r o o f . Suppose that S E G J H , K ) . Moreover, let X E B(Z2, H ) and Y E S(l,, K ) with llXl[ 5 1 and [[Yl[5 1. Form the sequence t := ((SXe,, Ye,)). Put PMx:= eMz for x € 4, where M is any finite set of natural numbers with card ( M ) 5 n. Since Y*SXPM E Gl(12,12), by the preceding lemma we have n
n
Therefore it follows from 13.7.7 that n
n
21 gk(t)2 2’ sk(S) 1
for n = 1, 2,
...
So we get ((SXen,Ye,)) E a and a((SXe,, Ye,)) 5 S,(S). Conversely, let S B(H, K ) such that the above condition is satisfied. Given E > 0, by 11.3.5 there are operators X E B(12,H ) and Y E B(&, K ) with l[Xi 5 1, IlYll 5 1, and s,(S)
2 (1
+
E)
(SXe,, Ye,)
€or n = 1,2, ...
Hence (s,(S))E a and ASE G,(H, K ) . Moreover, we have
S,(& = a(s,(S)) 5 (1
+
6 ) a((JXe,,
Ye,)).
15.4.4. As a consequence of 15.4.3 we obtain the Theorem. Let [a, a] be a mrmed sequence ideal with domination property. Then
[Go, S,] is a norrned operator ideal having the dominatim property, a8 well. Proof. Using
Sa(S) = SUP {a((#xe,, Yen)):IlXll I1, IIYII 5 I} we see that f& is a norm. To check the completenesswe consider an S,-Cauchy sequence of operators 8, E B,(H, K ) . Then there exists 8 E B(H, K ) such that S = [[.ii-lim8,. k
15. Operator Ideals on Hilbert Spaces
21 I
Let X E B(Z2, H ) and Y E 2(Z2,K ) with llxll 5 1 and llyll 2 1. Put tk := ((SkXe,, Y e , ) ) and t := ((SXe,, Ye,)). Given E > 0, we choose ko such that S,(sh - 8,) 5 E for h > k 2 ko.This means that a(tn - tk) 5 E for h > k 2 k,,. Hence (tk)is an a-&uchy sequence which obviousIy tends to the limit t. So we have t E a. Therefore S E Ba(H,K ) . Moreover, it follows from a(t - t k ) 5 E that Sa(S- 8,) 2 E for k 2 ko. Consequently S = Sa-limSk. Using D.3.3 it can easily be checked that [Go, S,] has the doniik
nation property.
15.4.5. Summarizing 15.4.2 and 15.4.4 we obtain the main result of this section. Theorem. The rules define a one-to-one correqondeme between all normed operator idmls with dorniwh property on the class of Hilbert spaces and all normed sequence ideals with domination property on the scalar field. Remark. Let us mention that minimality and maximality are carried over in both directions.
15.4.6. We now check the compatibility of the *-procedures for normed operator ideals and sequence ideals. Theorem 1. Let [a, a] be a nomzed sequertce ideal having the dOmination property. Then lea*,%*I = [e:, 8 3 Proof. I n the following we use criterion 10.3.6. Take the Schmidt factorizations L = X L o Y * Let S Go*@, K ) and L E S ( K , a). and Lo = X*LY, where Lo E 8(12, 4) is of the form Lo(&)= (&En) with (A,) E f. Since the maximal normed sequence ideal [a*, a*] has the domination property, we have ltrace (SL)I = Itrace (rSXL,Y*)I =
1:
(SXe,, Y e , ) 2,
- a*((SXe,, Ye,))a(&) 5 S,.(B)
0, wc choose i E 3 ( l )such t,hat
Proof. Given E
P u t P := 2; xr @ x i . Then i
Hence ]IS- SPll 5 E . So we l a v e shown that S is approximable. Consider a Schmidt. representation
s=
u p ; @ 2:j J
0. Then
such that ai
liSxi12 =
0 ;
l(x2.uj)j2 for all
EH.
5
J
Using Holder’s inequa1it.y with the exponents 2 / p and 2/(2 - p ) we get uiy =
uip
J
=
r~
5
(F /(xi,
Ui)/2)
2’ ap I ( X i , U j ) l P 1(5i,uj)l2-p
c (E u; I
!(Xi,u#)PA
J
( X 1(5i, U q o P ) ’ 2 J
I 2 r 2 ! ( P i ,.Ui)12 - I ( ?
j
)
‘
=
2 II8PillP. I
This proves that S E Gjp(H,K ) and
S,(S) =
(2upjl‘p 5 (2j!SZillPjl/P. I
J
< cy, md S E G J H , K ) . Then
L e m m a 2. Let 2 5 p
(T IIS~illP)1’P d
SpW)
f o r every orthonormal basis ( x i )of H . Proof. Since fl E Q(H, K ) , we can find a Schmidt representation
s = 2 up; J
vj
214
Part 3. Theory of Sequence Ideals
Hence
15.5.5. If p = 2, then the preceding lemmas yield a n important T h e o r e m . An operator S E 2 ( H , K ) belongs to G, i f and only i f (jlSxi/l)E I,(I) fw every (some) orthonormal basis (xi) of H .
Ir, this case, S,(S)
=
z
lSSi112 1'2.
( I
)
R e m a r k . Usually G2-operators are called Hilbert-Schmidt operators. The basic examples are operators S in L,(Q, p) possessing the form Sf = g with dL")=
J sb, B ) f ( B ) WB) >
s)
where the kernel s belongs t o L2(Qx Q, p
x p). Then
.
For further informations we refer t o [DUN], [GOH], [RIAU,], and several other Hilbert space textbooks.
15.6.6. Theorem. [G,, S,] i s a normed operator ideal.
Moreover, the norm S2 can be generated by the scalar product
(8,T):= X (&xi,T x j ) for S , T E G,(H, K ) , I
where ( x i )i s any orthonormal basis of H . Proof. The existence of (8,T) follows from
2 I ( S X i , TZi)j 5 (? I!SXijlZ)1'2 I
(T
I l W 2 ) 1 / ~= wf9
SAT).
Obviously the above equation defines a scalar product such that S,(S) = (S.S)ll'. R e m a r k . Since all components G , ( H , K ) are Hilbert spaces, we see that the S,] is self-adjoint. normed operator ideal [G2,
15. Operator Ideals on Hilbert Spaces
215
16.5.7. Using anot,lier method we prove a special case of 15.4.3. Theorem. Let 1 5 p < 00. An operator S E B(H, K ) belongs to 6, if a d o d y if ((XX%,Ye,)) E 1, for all 0pera.tor.sX E B&, H ) and Y E E(Zz,K). In th& case Sp(8) = SUP {lp((SXen, Yen)): IlXll 5 1, IlYIl 5 1). Proof. Suppose that the above condition is satisfied. Then by 15.2.1 we have S E @(E,9).Consequently there exists a Schmidt representation
s=
yi.
GjXf I
Without loss of generality we may assume that I = { 1,2,
...).Put
X(Ei):=rl: Eixi and Y(qi) := rl: qiyi. I
Then
~i
I
= (SXei, Yei). Hence (ai) E I,(I). This proves that 8 E G J H , K) and (SXei, Yei)).
S p ( 4 5 I,(
Conversely, let S E G,(H, K ) , X E B(&, H),and Y E So := Y*8X E Gp(Z2,Zp). Take a Schmidt representation
8,= 2 .ja$
B(&, K).
@ yi
I
such that ui 2 0. It follows from
2 ail(% en)12= rl: Gi I(% I I
I(%
e,)P
e,)lZ/P’
5 {$’ 0’ ~(zi, e n ) I z p {.$‘ I(zi, en)12)”*’
that
{F
~i/(xi,e n ) 1 2 r
en)12p.
5 {+‘ 0: ih,
Analogously we obtain
{+‘ ui I(Y~,
en)Iz)plz
{+’
0; I(yi, en)12ya
Noreover, t(&en, en)i 2
S Finally, we get
LI ‘4”I(si, e,)I ~i”’ Kyi, e,)I
k
{F
bi I(zi, en)12)llz
Ibi,
en)12y*
Then we have
216
Part 3. Theory of Sequence Ideals
15.5.8. We now formulate the main result of this section.
T h e o r e m . [GP, S,] is n n,ornied operator S e a l for p ;2 1 n?zd ideal for 0 < p -==c1.
CI
p-nornted operator
Proof. We know from 14.2.2 that [Gp, S,] is a quasi-nonned operator ideal. If p 2 1, then the triangle inequality follows from
S,(@ = slip (l,((SXe,, Ye,)):IIXII I 1, IIYII I I}. On the other hand, 18.5.1 and 15.5.2 iinply the assertion for 0
< p < 1.
15.5.9. The next. statement is also of great importance; cf. 14.2.3.
< p , q < CQ and l/r = 1/p + l/q.
Theorem. Lrt 0 [Gpj Spl 0
[Gq,
Sql
r z
Then
[Gr. S r l *
Proof. If S E G,(H, K ) , thcm wc consider a Schmidt representation
s = 2' a& 0yi I
such that (ai) E 1,(1)and ui 2 0. Put
S(&) :=
and
of:u&q
Y(qi):==
@'iliyi.
I
I
Then S E Gq(Z2(1), H ) and it follows froin
S q ( X )= that Hence
5.'
(7
4
d
llg
Y E Gp(Z2(I), K ) . Obviously we have 6 = Y X * . Moreover, and Sp( Y ) =
s, 0 S,(S) 5 ( c u; I 1 [Gr,
2 aT ( I
I)
= S,(S).
Srl E [Gpt S p l * [Gq, 891.
We now check the converse inclusion in four steps. By 15.2.5 i t is enough to consider operators acting in s fixed Hilbert space H . (1) Let S E G2(1$,13):\rid T f G,(H, H ) . Take a Schmidt representation
T =-:
2' T~X: '9y i Ja
such that t i 2 0. Choclse an orthonormal basis (xi)with i E I which is an extension of the orthonormal family (xi) with i E Io. Then Tzi = tiyi for i C I . and T x , : o for i @ I,. Consequently
+
where 1Jr = 1/52 l / q . Sinoc 0 < r 5 2, by 15.5.4 (Lemma l ) ,we have ST E G,(H, 11) S2(S)SJT). This proves that and S,(sT)
rB,, S,1
o
En,S,1 E 1 6 , S,]
with
llr = 1/2
+ l/q-
15. Operator Ideals on Hilbert Spaces
21 7
(2) According to 15.1.1, by (l),we obtain
[G,, S,] o [G2, S,] 2 [G?,S,]
l/r = l/p
with
+ 112.
(3) Suppose that S E G,(H, H ) and T E G g ( B ,If), uliere 2 5 p , q < my and 1 ‘r = lip f l/q. Then 1 5 1’ < co. Let X E 2(Z2,If) and Y E 2(ZZ,H ) such t h t X!1 5 1 and JJYI!5 1. By 15.5.4 (Lemma 2) we have
5 S,(S*Y) S,(TS) 5 SJS) S,(T). ”her? 15.5.7 implies SI’
E G,(H, H ) and S , ( S T ) 5 S,(S) S,(T). Hence
[Gp,S,] o [G,, S,] C= [G,, S,l with l / r == lip
-t-
l/q.2
5 p, g < w.
(4)To treat the general case we choose integers rn a i d n snch that l / p - 11.2 2 mi2 < lip and
l/q
-
112 5 n / 2 < l;q.
If 3/po:= lip - m/2 and l/qo := l/q - n/2, then 2 5 po, qo < S :G,(H, H ) arid 1’ E G,(H, H ) , there are factorizations
S 15
ASra.. . Slk90 a d
T
00.
Given
..
= llolll. T,,
ith So t G J H , H ) and
S,,(So) = S p ( S ) P ’ p a ,
To f G,,(H, H ) and
Sq6(Y0)= Sq(T)4’90,
Sh E G,(H, U ) arid
S,(S,)
= S,(S)p’*,
’r, E G,(H. E I )
S,(T,)
= Sq(T)Q‘z,
and
where h = 1, ..., m and k = 1,
and
ST
= (S,,,
- a *
...,n. Now
S,(SOl’o) T1
S,(ST) 5 S,(S,)
*. *
T,$) f G,(H. H )
S,(S,) S,.(SO)
- Sp(S)P/2+-’PP
it follows from ( I ) , (2), and (3) that
PlPO
S,@(TO) S,(T,) .’. S,(Y1n)
84 ( q q ” - “ .
q/Z+qlq,
= S,(S) Sq(2’).
This completes the proof.
15.5.10. Finally, we mention an iiiiiiiediate coiisequcnce of 12.2.5 and 12.3.1. T h e o r e m . Let 0
< p < m. Then G,
= @, on
the class of Hilbeit ~prrces.
Remark. Obviously the quasi-noriiis S, and E, do not coincide, however, they are eauivalent..
218
Part 3. Tlieorv of Seauence Ideals
15.6.
Extensions of Operator Ideals
15.6.1. In this section U, denotes some operator ideal on the class of Hilbert spaces. An operator ideal U defined on the class of all Banach spaces is called an extension of f& if U(H,9) = U,(H, K ) whenever H and K are Hilbert spaces. 15.6.2. An operator S E B(E,F ) belongs to the superior extension Urp if it holds BSX E 910(H,K ) for all X E B(H, E ) and B E O(P,K ) . 15.6.3. We have the simple
a,. Proof. Obviously UEupis an extension of a,. Now let Theorem. Urp is the largest extension of
bc an arbitrary extension. Suppose that S E %(E,F). If X E B(H, E ) and B E B(F, K ) , then it follows that BSX E U(H, K ) = %,(H, K ) . Hence S E U:up(E, F).This proves that E UFp.
15.6.4. An operator S c C ( E , F ) belongs to the inferior extension Up‘ if there exists a factorization S = YS,A, where A E B(E,H ) , S o €U,(H, K ) , and Y E B(K, 2;”). 15.6.5. Analogous to 15.6.3 we obtain the Theorem. i s the smallest extension of 8,. 16.6.6. An operator S E C ( E , F ) belongs to the right-superior extension sZgp if, given B E B(P, K ) , there exists a factorization RS = SoA, where A E O(E,H ) and So E Uo(H, K ) . 15.6.7. Theorem. U r p 6s an extension of Uo. Proof. Let S,, S, E Urp(E,F ) . Given B E O(F, K ) , we can find factorizations BSi = So+4i,where A , E B(E, H , ) and So, E %,(Hi, K ) . Let H := H , x H2 be the Cartesian product wit.h the Euclidian norm. Then
+
+
+
B(fJ1 S2) = (SOlQ1 So,&,) (JlAl J2A2). Now it follows from SolQ1 + S,Q, E U,(H, K ) that S, -+S, f Urp(E,F). This proves (OIl). Since (01,) is evident, must be an operator ideal, Obviously U%P(rr,K ) = U,(H, K). 15.6.8. An operator S E E(E, F) belongs to the right-inferior edension 9lF if there exists a factorization S = Y A , where A E B(E, H ) and Y E B(H, F ) such that BY E a@, R)for all B E B(F, K ) . 15.6.9. Theorem. i s an extension of a,. Proof. Let S,, S, E Up(E, F ) . There are factorizations Si = YiAi, where A( E O(E, Hi) and Yi E B(Hi, F ) such that BYi E %,(Hi, K ) for all B E B(P,K ) . Let €1 := H , X H z be the Cartesian product with the Euclidian norin. Then 8,
+ 8, +
=
(Y1Ql
+ Y?Q,)(JiA, +
J2-4)
+
and B(Y,Q, Y2Q2) E 910(H,K ) for all B E S ( F , K ) . Hence S, 8,E Up(E, F ) . This proves (Orl). Since (01,)is evident, U r n must be an operator ideal. Obviously Ut;’”(H,K ) = A o ( H ,K ) .
15.6.10. An operator S E f?(E,F ) belongs to the left-superior extension UzlUp if, given X E B(f.1, E ) , there cxists a factorization SX = YS,. where So E a0(H, K ) and Y E O(K,F).
15. Operator Ideals on Hilbert Spaces
219
15.6.11. Analogously to 15.6.7 we have the
Theorem.
atpis an extemhn of a0.
15.6.12. An operator i3 6 O(E,P) belongs to the left-inferior extension 2Ip if there exists a factorization S = Y A , where A E B(E,9)and Y E B(K,F ) such that A X E a0(H,R ) for all X E B(H, E). 15.6.13. The next result is analogous to 16.6.9.
Theorem.
is an eztension of 910.
15.6.14. Clearly the class of all extensions becomes something like a complete lattice with respect to the natural ordering. In the following diagram the arrows point from the smaller operator ideals a t the larger ones.
T Remark. The extensions conctructed above do not coincide in general. However, there are operator ideals on the class of Hilbert spaces admitting a unique extension. A trivial example is the ideal of finite operators. See also 18.7.9. 15.6.15. Using the technique of 8.4.9 we can prove the
Proposition. The operator ideals a?, Up, and
@f:
are injective.
15.6.16. Analogously we get the
Proposition. The operator ideals
arp,Cap,and @cfare mrjedive.
15.6.17. We mention without proof the following result.
Proposition. The operator ideal
2 completely symmetric.
15.6.18. We now formulate the open
Problem. Is the operator ideal
Urpcompletely symmetric?
Remark. If all extensions are restricted to the class of reflexive Banach spaces, then we have the following relations: (%rp)du.%l = a s u p 0
(Ul‘’P dual
T=
(%p)dusl
=
0
)
UruP 0
0
win
dual
)
(%?)dual
0
(gy)aual = ginf 0
’
= ulUP 0
=
alin
220
Part 3. Theory of Sequence Ideals
It can be shown that, the right-hand forinulns hold in general. The last equation is equivalent t,o the assertion of 15.6.17. 16.6.19. If I is any extension of
a,,
t.hen
ugl' = 8-10 a 0 9-1 at;'"= (80 U) 0 8-1 a""= sj 0 (80 9-1) !@f= 9 0
%;;UP = 8-1 0
(U 0 8)
%.,i= ll (8-1 0 U) 0 sj
u 8, 0
where 8 denotes the ideal of all Hilbert operators; cf. 6.6.1. If [a,, A,] is 8 quasinormed operator ideal on t,he class of Hilbert spaces, fhen the shove extensions become quasi-normecl operator ideals in a canonical way; cf. 7.1.2 and 7.2.2.
15.7.
Notes
J. W. CALKIN observed in 1941 that theic is
it one-to-one correspondence between operator ideals on the separable Hi1bci.t space and certain sets of non-increasing sequences of nonnegative numbers. See also [SAT2, p. 261. According to D. J. H. G-ARLIXG[2] it is possible to replace this so-called characteristic sets by sequence ideals. The operator ideals G p correspond[I]. ing to the classical sequenee ideals I, have been studied by J. VON x E U M A N N / R . SCHATTEN The case p = 1 was already treated in a paper of F. J. MURRAY/J. vos NEUMANR [l], and [l]. Full treatements for p = 2 one gets the ideal of Hilbert-Schmidt operators; cf. E. SCHXIDT of this theory may be found in the monographs [GOH, pp. 91-95], [RIN, pp. 75-1073 and [SAT,, pp. 29-43] as well as [DUN, pp. 10%- 11001. The problem of extending operator ideals from the class of Hilbert spaces to the class of Banach spaces has been extensively studied by [7], [lo], [21]. the author; cf. A. PIETSCE
Recommendations for further reading:
A. BROWN/C. PEAP.CY/N. SALIEAS [l], CII. C>ONSTAXTIN [l], I. C. GOHBERG/M.G. BEJN (11, J. R. HOLUB[4], [S]. H. J. JUKEK[l]. A. S. XARKUS [l], K. MAURIX[l], C. A. NCCABTHY[l], C. MERRECI/PHAXTEE LAI[l],D. MORRIS/N. SAUNAS [l],F. J. MWRBAT/J. voa NEUMANN [l], w.OOBTESBRIh-K [l], -4.PEECZP~~SKI [4], -4.PIETSCH [121, D. J. VOX RIEMSDIJK [I], s. SALINAS[l], [2], [3], 13. SPRATTEE [l], K. TSUJI[i]. See also the bibliographical notes of Chapter 28.
16.
Family Ideals on Banach Spaces
In the following we introduce the concept of a family ideal on the class of arbitrary Banach spaces. Most results obtained for scalar sequence ideals could be carried over to this general situation. However, we only f0rmulat.ethe definitions and a few elementary statements.
The main purpose of this chapter is to treat some basic examples which we need later on. The ideals of absolutely and weakly p-sumniable families are of special importance. Furthermore, we consider ideals of so-called mixed (s,p)-summable families.
16.1.
Family Ideals
16.1.1. Let us recall that I denotes the class of all bounded families in arbitrary Banach spaces. A family deal
is a subclass of I such that the components
a(E, I) := a n I(E, I) satisfy the following conditions:
(Fb) e, E a, where eo = (1)is the single unit family. (FI,) It follows from "E, z,E a(E,I)that x1+ x2 E a(E,I). (FI,) If s E l ( I )and z E a(E, I),then m E a(E, I). (F13)Let z be any one-to-one map from I, into I. Then J,x E a(E, I) for all 3~ E a(E, I,) and Q,x E a(E,I,) for all z E a(E, I). (FI,) If S E 2(E7F ) and x E a(E, I),then Sx E o(F, I). Remark. Family ideals will be denoted by bold small Gothic letters. Remark. For the components on the index set M = (1,2,. ..] we shall simply write a(E).
16.1.2. Proposition. Let a be a family ideal. Then all components a(E, I)are linear qme-s.
16.1.3. Let be a family ideal. A map a from a into W+is called a quasi-normif the following conditions are satisfied :
(QFI,)a(eo)= 1, where e, = (1) is the single unit family. (&PIl) There exists a constant x 2 1 such that
+
a(xl
8,)
+ a(xz)l for
5 x[a(z,)
a,,xzE a@, I).
(QFI,) If s E l(I)and a E a(E, I ) , then a(sx) I ((slla@). (QFI,)Let n be any one-to-one map from I, into I. Then a(J,z) a(z) for all z E u(E,I ) . x E a(E, I,) and a(&#) (QFI,) If S E 8 ( E ,3') and z E a(E, I),then a(&) 5 llSll a@).
a ( a ) for all
324
Part 4. Basic Examples of Operator Ideals
Remark. Quasi-norms on family ideals will be denoted by bold sinall Roniaii letters.
16.1.4. Proposition. Let a be a family ideal zuithaquasi-norm a. 1’henallcon~ponent.s a(E,I ) are linear topological Hausdorff spaces. 16.1.5. A quasi-nornied family ideal [a, a] is a family ideal a with a quasi-norni a such that all linear topological Hausdorff spaces a(E, I)are complete. 16.1.6. Proposition. Let [a, a] be a qunsi-normed family ideal. Then 1 1 ~ 1 15 a(x) E a.
for all x
16.1.7. A quasi-norm a on the family ideal a is said to be a p-norm (0 < p 5 1) if the p-triangle inequality holds:
a(xl
If p
==
+ a,), 2 a(al)p + a(xz)p for 3cl,x2 E a(&‘,I).
1, then a is simply called a n m w .
16.1.8. A p-normed family ideal [a, a] is a family ideal a with a p-noriii a such that all linear topological Hausdorff spaces a(E, I) are complete. If p = 1, then [a, a] is said to be a norm& family ideal. 16.1.9. Let [a, a] and [b, b] be quasi-normed family ideals. Then [a, a] E lb, B] means that a & b and a(a) 2 b(x)for all x E a.
16.2.
Absolutely p-Summable Families
16.2.1. Let 0 < p 5 00. A family x = (xi), where xi lutely p-wnamable if (ilxi[l)E & ( I ) .We put
E for i E I, is called abso-
l,(xi) := lp(ilxill). The class of all absolutely p-summable families is denoted by I,. Remark. Obviously [Irn,1-1 = [I,
11.11].
16.2.2. Theorem. [I,, l,] 12 a normed family ideal for p 2 1 nnd a p-normed family idealfor 0 < p < 1. 16.2.3. Proposit,ion. Let 0 < p1 5 p, 5 00. Then
u,, lP,l G [I,,, 1PJ 16.3.
Weakly p-Summable Families
16.3.1. Let 0 < p 5 00. A family z = (xi), where xi E E for i E I, is called w ~ U Y p-summable if ((ai, a)) E I,(l) whenever a E E’. We put
w,(q)
:= sup {ip((xi,a ) ) :Ilall 5 11.
The class of all weakly p-suminable sequences is denoted by t~,.
16. Family Ideals on Banach Spaces
228
Remark. The existence of wp(zi)follows from the closed graph theorem or the principle of uniform boundedness. Remark. Obviously [w,,
w,l
:= [I,
11.111.
16.3.2. Theorem. [mP,wp] is a normed famity ideal for p < p < 1.
>= 1
and u p-normed
f.miZy deal for 0
16.3.3. Proposition. Lei 0 [WP,, WP,]
16.4.
< p , 5 p2 =( 00. Then
s b p , , WP.1.
Mixed (s,p)-Snmmable Families
+
16.4.1. Let 0 < p 2 s CQ and determine r by 1/r 11s = lip. A family x = (xi), where xi E E for i E I , is called mixed (s, p ) - s u m d l e if it can be written in the form z,= zjxtwith (ti) E l,(I)and (z!) E m,(E, I ) .
Put m(s,p)(xi) := inf & ( t i ) w&%)I,
where the infimum is taken over all possible factorizations.
The class of all mixed (8, p)-summable families is denoted by m(s,p). Remark. I n the boundary cases we obviously have [ r n ( p , p , , In(p,p)l
= [Wp, Wpl
and
[ t t f m , p , , m,m,p,I
= [ 4 7 > 1Pl.
16.4.2. Theorem. [tq,,,, m(s,p)] , is a normed family ideal for p 2 I and a p-normed family ideal for 0 < p < 1. Proof. We only mention that the p-triangle inequality follows from the next criterion.
m(s,p)]can be considered in an obvious way, see 7 . 1 2 ,
Remark. Clearly as the product of [lr, l,] and
[a,, w,].
16.4.3. We now give an important characterization. (8,
Theorem. Let 0 < p < s < 00. A family (xi), where xi E E for i E I , ia mixed p)-summable if cind d y if
((
\(xi,u)ls dp(a))1'8)E $,(I) whenever p E JC'(UEp).
UE'
Here we suppose that UEt is equipped with the weak E-tOpdO[y?J.
1%this case, m(s,p)(zi) = SUP
{[+(rjLJ .>I") I I(xi,
dp(a)
:P E
1-
JV-~UE,)
Proof. Let (x,)satisfy the above condition. Suppose that there are pn E W ( U E 3 ) with
15 Pietsch, Overator
226
Part 4. Basic Examples of Operator Ideals m
By setting p := 2 2-n,u,,we obtain ,u E W (UEs)such that 1
This contradiction shows the existence of
Put u := r / p a n d v := slp. Then l / u compact convex subset
K:={
+ l l v = 1. We now
consider the weakly
(ti):z E r 5 op and t i 2 0
1
I
of &(I). Observe that the equation
@(ti) :=
i
+
(ti
E)-O
J [(xi, UP;*
.>I" d ~ a, )
where ,u E W(U,,), E > 0, and i E 3(I),defines EL continuous convex function @ on K . Take the special family (ti) with
Then (ti)E K and @(ti)5 o p . Since the collection 3 of all functions @ obtained in this way is concave, by E.4.2 we can find (6;) E K such that @(&;) 5 o p for all @ € ,F simultaneously. In particular, let S(a)be the Dirac measure at the point a. Then we obtain 2 ($ .)-" ](xi,a y (1op
+-
t
for E > 0, jjalj 5 1, and i E 3(I). If xi 0, it follows that 5: 0. Hence we can put zi := ]Epll/Pand x: := t;'xi. In case xi = o let. ti := 0 and x! := 0. Then xi = zixf.Moreover, we have
+
+
(q
lril')l/r
d (2Igp)l.
5 f+
I
and
for i E % ( I ) and l]u/l 5 1. Hence lr(ri)w,(xp) S cr. This proves the necessity of the above condition.
Conversely, suppose that (xi)is niixed (s,p)-summable. Take any factorization
xi = TjZp with (zi) E &(I)and (xf)E m,(E, I ) . Applying Holder's inequality we obtain
16. Family Ideals on Banach Spaces
227
I l,(Zi) w,(z;) whenever p E W(U,.). This prores the sufficiency of the above condition.
16.4.4. The next inclusion follows immediately from 16.4.3. However, there is also a n eleiiientary proof; cf. 17.1.4.
16.5.
Notes
The definition of 8 family ideal seems to bc given in this monograph for the first time. The special ideals of weakly and absolutely p-summable families have been investigated by several [5]- The only non-straightauthors. I n particular, we refer the reader to A. GROTHENDIECK forward result of this chapter is a powerful characterization of mixed (s,p)-summable families the proof of which is mainly taken from B. XAUREY [2]. See also [SEN,,exp. 151.
Recommendations for further reading:
H. APIOLA[l], M.E. MUNROE[l], 51. S. ~ K L C P R[~ l],I LD. RUTOVITZ [l].
17.
Absolutely Summing Operators
This chapter is devoted to the theory of absolutely summing operators first defined by A. GROTHENDIECK in 1955.
.'r '1r
We deal with the following operator ideals :
%.p)
Pr
absolutely (r, p , q)-summing operators,
absolutely (7, p)-summing operators, (p, q)-dominated operators,
absolutely r-summing operators,
%
p-dominated operators.
The basic concept is that of an absolutely r-summing operator due to the author (1966). As the main tool we use a characterization of those operators by means of some fundamental inequalities. The definition of (11, q)-dominated operators goes back to S. KWAPIE& who also proved the formula B(p,qb = @:"&' o !&,.
17.1.
Absolutely (r,p , q)-Summing Operators
17.1.1. Let 0 < r , p , q co and 1/r 2 l/p + l/q. An operator S E E(E, F ) is called absolzltely (7, p , q)-summingif there is a constant u 2 0 such that lr((flsi9
bi)) 5 Q wp(si)wq(bi)
for all finite families of elements xl, We put
P(r,p,ql(~) := inf
...,x,
E E and functionals b,,
...,b,
€ F'.
(T
The class of all absolutely (7, p , @-summingoperators is denot,edby ?&r,p,qb.
17.1.2. The proof of the following result is straightforward; cf. 6.5.2. Theorem. [!$(rmp,q), P(r,p,qJis a nornaed operator ideal for i d f dfW 0 < 7 < 1.
7
2 1 and an 7-nom7d
opWd07
17.1.3. We now state a generalization of 8.7.8. Theorem. T?Mquasi-wmtd operator ideal [?&r,p,qb, P(r,,,,qJ-is mccxim~l.
229
17. Absolutely Summing Operators
Conjecture.
@(r,p.q)
is proper if and only if 1 / r > l / p
+- l / q - 112.
Remark. I n 17.2.7 we will show that the answer is affirmative for p (1 =
00.
= 00
or
230
Part 4. Basic Examples of Operator Tdcnls
17.2.
Absolutely (r,p )-Summing Operators
17.2.1. Let 0 < p 5 r 5 00. An operator S E O(E, F ) is called absolutely ( r , p ) summi?q if it belongs to the quasi-normed ideal [(P(r,p)>P(r,p)l:= [ ( P v , p , o o ) * P(r,p,oo)l-
17.2.2. We now give a more direct characterization. P r o p o s i t i o n . An operator S E 2 ( E , F ) is absolutely (r.p)-summing if and only if there .is a constant cr 2 0 such that
l,(SXi) 5 CT WJXj) for all finite families of elements xl,...,x, f E.
In th& case, P(r,p)(S) = inf u. Proof. The sufficiency of the above condition follows iniinediately from
lA(Sxi9 b i ) ) 5 MllB4l IIbill) 2 1ABXi) ww(bi) 5 wp(xi) wea(bi)*
To prove the necessity, given xl, ...,x,,, f E,we choose b,, @xi, bi) = IISxill.Then lr(SZi)
= lr({S~i> 6,))
...,b,,, E U,. such that
5 P(r.p)(S)Wp(zi)wm(bi) 5 P(r,p)(S)mrp(ai)*
R e m a r k . The preceding criterion shows that the absolutely (1,l)-summing operators coincide with the absolutely summing operators introduced in 6.5.1.
17.2.3. The phrase “absolutely (r,p)-summing” is derived from the following property. P r o p o s i t i o n . An operator X E B(E, F ) aS absolutely ( r , p)-summing i f m d only i f every weakly y-summable sequence i s mapped into a n absolutely r-mmmable sequence. Proof. The necessity of the above condition follows by applying 17.2.2 t o the finite subfamilies of (xi)E w p ( E ) . Conversely, if every weaklyp-summable sequence (xi)is transformed into an absolutely r-summable sequence (&‘xi), then we get a closed linear map from mP(E)into I@). Hence, there exists a constant u 2 0 such that lr(Sxi)I;c r ~ ~ ( x ifor ) all (xi) E wp(E).I n particular, this estimate holds for finite families.
17.2.4. The next statement is a consequence of 17.2.2. T h e o r em. The quasi-normed operator ideal
[?&r,p,,
Y,,,,,] is injective.
17.2.6. It follows from 17.1.3 that the quasi-normed ideal of absolutely (r,p)-sunlwing operators is maximal and therefore ultrastcable;cf. 8.8.7 (remark). For a direct proof we refer to 8.8.11. Proposition. The quasi-normed operator ideal
[Ti(,,,), P,,,,] is ultrastable.
231
17. Absolutely Summing Opcrators
17.2.6. As a special case of 17.1.4 we have the Froposition. Let rl 5 r2, p1 5 p 2 , and lipl - l/rl [ P ( r 1 , p 1 ) 9Y(r,,p,)l
1/p2 - l/r2. Then
-
[ P ( r 2 , p , ) j P(r2,ps)l
17.2.7. Theorem. The operator ideal (P(r,p) is proper if and only if l/p - I / r < p 2 r < 00.
< 112
and 0
Proof. It will be shown in 22.1.12 that the identity map of Z2 belongs to for 1/p - l/r 2 1/2. Moreover, we have I p ( m , p ) = 2 for all p
@(r,p)
> 0.
Suppose that E is an infinite dimensional Banach space. Given a natural number 12, hy 33.2.1 we can find x,, . ., 2, E such that IIzl/j = -.. = llz,Jl = 1 and w2(zi) 5 2. So ?@(r,2) is proper for 2 5 r < 00. Clearly !&r,p) & ?#(7n2) for 2 5 p 5 r < 00. for 0 < I, 5 2 and I/p - 1/r = 1/2 - l/ro. Hence the Noreover, ?@(r,p) E operator ideal ?@(T,P, with 0 < p 2 r < 00 and l/p - 1/r < 1/2 is proper.
.
L7.2.8. Let us define the space ideal P(7,P)
:= Space (%*p))*
In particular, a Banach space is said to have the Orlicz property if it belongs to
Po,,).
17.2.9. We now give an interesting example which has bcen investipted by
s. KwAxmlil/A.PEwzP6sKI [11.
Proposition. The operator S E B(E,, co) defined by
(T t i ) a3
S(&) :=
is absolutely (r, l)-sumniiny for 1 < r 5 00. Remark. It can be shown that the operator S defined above is absolutely (r,p)suinming even for 0 < p < r 5 bo. On the other hand, we have S 6 ?D(Z,, co). Hence ?&r,p) $ ?D. However, it will be proved in 17.3.12 that !@(r,r) c ?D for 0 < r < 00. 17.2.10. Pinally, we mention a trivia.1,but useful Proposition. Suppose that F 13 a n intermediate space of (F,,, F,) possesSing J-type 6. Let 0 < p 5 ro, rl 5 00 an.d l/r := ( I - 6)/ro i3/rl. If S E 2(E,FA), then S E ?&,o,p)(E, F,) and 8 E P(rl,p)(E, F,) imply S E ?@(r,p)(E, F ) . Moreover,
+
17.3.
Absolutely r-Summing Operators
17.3.1. Let 0 < r 5 00. An operator S E 2(E, F ) is called absolutely r-summing if it beloiigs to the quasi-normed ideal
Pr1 := [V(7,r,m), P(r.r.oo)I* I l e u ~ a r k .Obviously [?&,P,] = [!&r.r), P(r.r)land [Pm. P,] = [e, 11.111I3377
17.3.2. We now prove the fundamental characterization of absolutely r-summing opcrators.
232
Part 4. Basic Examples of Operator Ideals
Theorem. Let 0
< r < 00.
An operator S E a ( E , F ) is absolutely r-summing if a conatant cr >= 0 and a probability p c W(Ur) such that
and ody if there ezc&
llfwl 5
fJ
{ J I(%, .)I'
1
4-44
for xE8'. Here we suppose that UEcis equipped with the weak 3 - t q o l o g y . Uh'?
I n tlink m e
P,(S) = inf u. Proof. The sufficiency of the above condition is evident, since for xl, ..., x, E E we have
ju
{ J z'I+.;, UE
a)l7
a&)}
1lr
I B Wr(2i).
1
Conversely, let S E @,(El3') and put c := P,(S). Take C(UEp)'equipped with the weak C(ZT,,)-topology.Then W(U,,) is a compact convex subset. For any finite family of elements xl, ..,x, E E the equation
.
defines a continuous convex function @ on W(UEp). Choose a, E U p such that
If S(a,) denotes the Dirac measure at the point a,, then we have m
@(&%))
=
z [11~41-7
fJr
l(zi, ao)lr]= Ir(8xi)r-
5 0.
1
Since the collection 9 of all functions @ obtained in this way is concave, by E.4.2 there is po E W (U E j such ) that @(po)5 0 for all @ E .Fsimultaneously. In particular, if 0 is generated by the single family (x),it follows that ll8Zll' - fJr
j1(~,41r &,(a)
I0.
UEI
This completes the proof. Remark. Observe that there exists po E W(U,.) for which the best possible value of c, namely P,(S), is attained.
17.3.3. For operators actringon a Banach space C ( K )the preceding crit,erioncan be stated in a simpler form. The proof is completely analogous. Theorem. Let 0 < r < 00 and let K be a compact Hausdorff space. A n operator 9 E Q(C(K),F ) is absolutely r-summing if an.d only if there exist a constant c 2 0 and
17. Absolutelv Summing Operators 18
233
probability p E W ( K )suc?~that
ilsfll5 0 { J If(oJ)lrdp(a)}l’r
for ~ Z fZ E c ( K ) .
K
I n this cme,
P,(Is) = inf
0.
17.3.4. Let p be a probability on the compact Hausdorff space K. If 1 2 r then the canonical map from C ( K )into L,(K, p ) is denoted by J,.
< 00,
17.3.5. Proposition. Let 1 5 r < 00 and let K be a compact Hnusdorff space. An operator S E B(C(K),F ) .iS abdutely r-summing i f and d y i f there ex138 a commufative diaqrnm
cuith s m e probability p 5 W ( K )and Y E B(L,(K,p), F ) .
In this case, P,(S) = inf !lYll, where the infimum is taken over all possible factorizations. Proof. We see from 17.3.3 that J , E p,(C(K),L,(K, p)) and P,(J,) 2 1. Hence S = Y J , E ?&(C(K), F ) and P,(X) 5 IIY11. This proves the sufficiency of the above condition. Conversely, let S @,(C(K),P).Then we choose p E W ( K )such that IlSfll 5 G
{
l f ( W ) I r dp(o))”‘
for all f E C ( K ) .
K
Consequently, the operator Yo: Jrf + Sf extends to Y f B(L,(K,p),F ) with ilY!I 5 6.Moreover, we have S = YJ,.
17.3.6. The next statement is also along the lines of the preceding results. Proposition. Let 1 5 r
< 00
and suppose that F has the metric extension property.
.4n operador 8 E 2 ( E , 3)i s wb.ssolv,telyr-summing if and only if there exists a eominutative
diagram
E A 4
S
,P
il Y I
834
Part 4. Basic Examples of Operator Ideals
~2~i.j~ some conipact Hausdorff *spaceK , a probability p E W ( K ) ,A E B(E,C ( K ) ) ,a d Y E g(Lr(K, F ) . In
this case,
P,(S) = inf Ilyil IIAli, where the infimum is taken over all possible factorizations. P r o o f . The sufficiency of the above condition follows immediately froin 17.3.5. I,& S E !&(E, F ) and piit G- := PJS). Then there is a probability ,u E W (UBl)such that
By setting Ax :=f,, where /,(a) := (x,a), we obtain a n operator A E 2 ( E , C ( U p ) ) with jiAil = 1. Let J , denote the canonical map from C(UE,)into L,(U,,, p). Then Yo(Jrf,):= Sx defines an operator Yofrom M(J,A) into F. Hence, we can find an uxtexsion Y E 2(L,(UE,), F ) such that S = YJ,A and IlYIl = IIYoil 5 G-.This completes the proof.
17.3.7. If r = 9, then the preceding criterion is tnic for arbitrary Banach spaces. P r o p o s i t i o n . An. opemtor S E tltere exists LC commutatiue dingram
B(E,F )
i s Gbsolutely 2-summing i f and ordy ij
with some compact Hausdorff space K , a probability ,u E W ( K ) ,A 6 G(E, C ( K ) ) ,c u d y i- 2(L2(K,p),F ) .
TR this case, P,(S) = inf [IYIlIIAli w?here the infimum .is taken over all pas.&le factorizations. Proof. Suppose that S E b z ( E ,F ) and use the saiiie constiriction as in the previous proof. Obviously Y o oxtends continuously to M ( J 2 A ) .P u t Y := YoQ,where Q is the orthogonal surjection from L2(UE,,p ) onto X ( J , A ) . Then we have S = YJzA and IlYll = IlYoil 5 G-.
17.3.5. We now treat n further example. p ) be a memure space. If s E Lr(f2,p), P r o p o s i t i o n . Let 1 5 r < oc, and let (B, then Sf:= sf detines an operator S E ?&(GJB, p), L,(P, p ) ) with P,(S) = llSll = &sllr.
17. Absolutely Summing Operators
235
P r o o f . Given f l , ...,f, E L,(Q, p ) , for every choiceof al,...)L Y , E X there exists a y-null set N ( a i ) with
Hence we can find a y-null set fl such that (*) holds for all rationals al,..., a, E 2 and all w 6 N simultaneously. Then, by continuity, the same estimate is true for all al,...,a, E X . Consequentlv
Finally, we get
17.3.9. As a special case of 17.1.4 or directly from the criterion 17.3.2 we have the
P r o p o s i t i o n . If rl 5 r2, then [prl, P,,] & [p,,,P,J. R e m a r k . The canonical map JrIfrom CEO, 11 into L,I[O, 11 shows that the inclusion l , c ?$,, * is strict for rl < r2 and 1 2 r, 5 00. On the other hand, it will be proved in 21.2.11 that all operator ideals ?J3, with 0 < r < 1 coincide.
17.3.10. Let us recall that 8 denotes the ideal of coiiipletely continuous operators. P r o p o s i t i o n . If 0 < r
< 00,
then?, cB.
Proof. Let 8 E ?$,(E, F ) . By 17.3.2 we have llh'x/\5 0
1l(x, a)]
p(
{US?
lIr
"')
for all x E E .
Hence, given a weak zero sequence (x,,),Lebesgue's theorem of bounded convergence implies lim IIh'xn]1= 0. n
Clearly, the identity map of ll is completely continuous, but not absolutely r-sumnling.
17.3.12. Let us recall that ?Eldenotes the ideal of weakly compact operators. Proposition. I f 1 < r
< 00. then [Q,, P,] = [m, 111.11 o [Vr, P,].
Proof. Let S E ?$,(E, F). As in the proof of 17.3.6 we define the operator Yo by Yo(J,f8) := Sx. It may happen that Yo cannot be extended to all of L,(UEp, p). However, there is an extension Y to Fo := M ( J , A ) .Since 1 < r < M, we see that Fo is reflexive. Moreover, the operator X := J,A transforms E into Fo and is absolutely r-surnming. This completes the proof. 17.3.12. The next result follows from 17.3.9 and 17.3.11.
P r o p o s i t i o n . If 0 < T
c 00,
then !& c m.
230
Part 4. Basic Examples of Operator Ideals
17.4.
c2,, y)-Dominated Operators
17.4.1. Let 0 < p , q 2 00. An operator S E B(E, F ) is called ( p , q)-dominated if it belongs t o the quasi-nornied ideal
[P(P4)’=(,,)I where l / r := IIp
:= [ P ( r . P , q P p(r,P,q)l
+ l/q.
17.4.2. We now generalize the criterion given in 17.3.2. Theorem. Let 0 < p , q < co. An operator S E B(E, F ) is (p,q)-dominatedi f and only i f there exist a constant CT 2 0 us well as probabilities p E W (U E , )and v E W (Up,,) .such tlmt
for nll x E E and b E F’. Here we suppme that UE,and UF#,are equipped with the weak E-topology and F’-topology, respectively. In
this case, D(p,q)(S) = inf
CT.
Proof. The snfficiency of the above condition is evident, since for .zl,..., x, and b,, ..., b, E F’ we have
E
5 C T W ~ ( WX q~()b i ) . Conversely, let S E 9 ( p , q ) ( EF) ) and put CT :== D ( P , q ) ( STake ). [C(UB,)x c(uF?,)]’ equipped with the weak C(U,,) X C(0’,*l)-topology. Then W(U,,) X W(UFr,)is a compact convex subset. For any finite families of elementjs xl, ..., x, E E and functionals bl) ...)b,, 5 F’ the equation
defines a continuous convex function @ on W (UE,) x W (UFTp). Choose a. E UE, and &‘ E Up,, such that and
j3 := wq(bj)=
17. Absolutely Suinming Operators
If a(%) and 6(y,”) denotes the Dirac measure at the point, a. and then we have @(a(ao), a(&))
237
&, respectively,
07
I(SZi, bj)“ - -
=
P
P
ar r
1 r
5 - lr((8Zi3hi))‘ - - (FB)’ 1 r
= - [l,((sxi,
bi))’
- ( m p ( z & ) wQ(bi))‘]
5
Since the collection .F of all functions @ obtained in this way is concave, by E.4.2 vo) 5 0 for all 0 E 3 there are p,, E W(UBt)and yo W ( U p ) such that @b0, simultaneously. I n particular, if @ is generated by the single families (x)and (b), it follows that 1
- l(Sr, b)j‘
r
-
“1
s
,(x,.)pJ dpo(n) - d 4
2,
l{y”, b p dv,(y”)
5 0.
Ufi.,
UE’
Finally, we put 112
:=
{ul
I(%, ~ > Idpo(a))l’p P
and n := {UJ, KY”,
w dvO(~’’)
Then
This completes the proof. 17.4.3. Theorem. Let 1 5 p , q [%(p,q), D ( p , g ) ]
< 00.
[(P:ual, p y 1
Then
p~l-
Proof. Suppose that S E g ( E , F ) can be written as S = Y A with A E P p ( E ,M ) and Y E pt“’(M, F).Given xl,...,x, E and b,, ..., b, E F’,it follows that L((SZi, bi)) = lr((Ari, Y’bi))5 1r(llA~illIIY’bill)
5 Pp(A) wp(zi) pQ(y‘)wQ(bi)* Hence S E %(p,Q)(E, F ) and D(p.Q)(S)5 P y ( Y ) P,(A). - l p ( A x i ) lq(y’bi)
Conversely, let S E % ( p , g ) ( E P ,) and put c := D(,,q)(8).By 1 7 . 4 % there are probabilities p € UB,)and Y E W (UFrp) such that
w(
238
Part 4. Basic Examples of Operator Ideals
for all x E E and b t P'. Let /,(a) := (x, a). Form the subspace M of Lp(UEr, p) spanned by all f, with x E E. By setting Ax :==fi we obtain an operator A f V P ( E ,M ) such that P,(A) 5 1. On the other hand, there exists an operator Y f !2(N,P)with Yf, = Sx for all 5 E. It follows from
IIY'b1/ = SUP {I(/,
Y ' b ) ] :lifzil 5 1) = SUlJ { ~ ( S X b)j: , llfiil 2 1)
that Y' E Vq(P',M') and P4(Y')
s CT.Moreover, X = Y A . This coinpletes the proof.
R e m a r k . Let us mention that
[P@X+ D,,,,,]
=
and
[ V p , Ppl
[P(,,?), D(,,,,I
= [P:ual, py1-
17.4.4. As a special case of 17.1.4 or directly froni the criterion 17.4.2 we have the P r o p o s i t i o n . If p , 5 p 2 and q1 5 q2, then
[P(P,4,),D(P,,q,)l sz
[%J,,r7,),
D(Pz.qs)l*
17.4.5. We now state a counterpart of the preceding inclusion. P r o p o s i t i o n . If 0
< p , q < 00,
then P(p,a) G
P(2,Z).
+
Proof. By 17.4.4 we may suppose that 11. := 1/23 l / q 5 1. Let US recall that d m denotes tho set of all vectors e = (cl, ..., E,) with c i = f 1. If xl, ..., x, E E and b,, ..., b, E P',then we put m
m
1
1
x e := 2' cixi and be := 2 cibi.
Using Khintchine's ineqnality we get
Hence wp(xe)5 Cp2m'PW2(xi). I n the same way we niay check that Wq(be)I Cq2m/4W2(bi).
P ) it follows froni Finally, if S E p(p,q,(E, m
17. Absolutely Slimming Operators
239
17.4.6. By 17.4.4 and 17.4.5 we have the P r o p o s i t i o n . If 2 5 p , g < 00, then P(p,q)= P(2,2).
17.47. Let 1 p 5 to the nornted ideal
00.
An operator S E f ! (E , P )is called p-dominaled if it belongs
[a,, Dpl := [*(l,P,P% R e m a r k . Obviously
17.5.
P(l,P.P*)I*
[a,, Dp] =
D(p,pe)].
Special Absolutely ( r ,p , q)-Summing Operators
17.5.1. We begin this section with the statement that for certain exponents all operators are absolutely ( r , p , g)-sumniing.
[a,
P r o p o s i t i o n . [*(l*l*l),P(1.1.1)I= 11.111Proof. Let xl,..., xm E E and a,, ..., a, E', where E is any Banach space. As in the proof of 17.4.5 we put
zcixi na
2, :=
I)(
and
a, := 2'
1
E~CZ~.
1
Then llzell-I ml(xi) and l\aell 5 wl(ai).It follows from m
2 (xi, ai) = 2-m2 (xe, a,) 6"
1
that
17.5.2. T h e o r e m . On the class of Hilbert spaces [Ip(r,z,z), P(r.z.~)l = [Gr, Srl for all 1 5 7
< 00.
Proof, Suppose that S E &(H, K ) , where H and K are Hilbert spaces. Given ...,x,,, H and yl, ..., y, E K , we put
~ 1 ,
m
X(Ei):= 2 tixi
m
and
Y(qi):= 2 qiyi.
1
1
Obviously X E 2(&,H ) and Y E 2(Z2,K ) . Moreover, IIXll SO, by 15.5.7, we have lr((szi,
~ i ) ) = 1r((SXei, Yei))2
= w2(xi)and
llyll = w&$.
Sr(s) w Z ( ~ i' ()7 ~ 2 ( ~ i ) .
Hence f i E rP(r,,,z)(a, K ) and P(r,z,z)(s) I Sr(fi)* Clearly, the identity map of l2 is not absolutely ( r , 2,2)-summing. Therefore the restriction of !&r,z,z) to the class of all Hilbert spaces is proper. We see from 15.2.2
240
Part 4. Basic Examples of Operator Ideals
that every operator S E !&,,,,,(H, 9)is approximable. Let
8=
qxi*@ yi I
be a Schmidt representation. Suppose that i E g(I).Then for all finite subfamilies ( x i ) and (yi) with i E i we get
(7 IQi
)
/ I l'r
-
4
1 (8% ?/d) 2 P,,.z,z,(4.
Consequently (oi)E & ( I )and l,(ai) 5 P,,,,,,,(S). This proves that ASE Br(H,K ) and S 9 w 5 P(r.z.z)(@. Remark. It can easily be shown that ?&r,z,z,, as an operator ideal on the class of all Banach spaces, is the superior extension of 6,; cf. 15.6.2.
17.6.
Relationships to Other Operator Ideals
17.6.1. Since p2 contains operators which are not approximable, we have QZ$ U2. Moreover, the converse inclusion also fails. Proposition. U24 !&.
Proof. Put o,,:= [ n log (n + 1)l-l and let S E 2(Z2,Zl) be the diagonal operator generated by (o,,). Then, according to 11.11.4 (theorem), we have 00
c c k
m o o
cU n ( S P = c Lp a; c 1
n=l k = n
k-1
00
0; =
=
n-1
c kI$
< 00.
1
Therefore S 6 ?&. On the other hand, it follows from (a,) absolutely 2-summing.
4 1,
that S cannot be
Remark. As observed by H.KONIG[4] we have a somewhat weaker inclusion, c !&, where 21(z,,) denotes the operator ideal generated by the approxinamely mation numbers and the Lorentz sequence ideal l(a,l).
17. Absolutely Summing Operators
241
17.6.2. Kext we state a n immediate consequence of 17.3.7. Theorem. [!&, P2]S [$j, HI.
17.63. P r o p o s i t i o n . [(p,, P2]o [$,HI G
[W,,A,].
P r o o f . Obviously it is enough to show that, for every Hilbert space H , it follows from S E '&(TI, F ) that S E %,(El, P ) and A2(S) 2 P2(S).For this purpose, according to 17.3.7, we consider a factorization
.t
A
Y c
such that / / A ,=( / 1 and /IY/l4 P,(S). We see from 17.5.3 that J2A E !&(H, L2(K,p ) ) implies J2A E U,(H, L,(K, p)) and A,(J,A) 5 1. Therefore Y = Y J J E W 2 ( H ,3') and &(S) 5 P,(S).
17.6.4. The precediiig results yield the Theorem.
[p,,PJ 5 [a,,
A,].
Reinark. Using the identity map from for O < p < 2 .
Zz into Zy we can easily show that
?&$%lP
17.7.
Notes
The concept of a n absolutely 1-summing operator (application senil-integral B droite) was introduced in the fundamental memoir [GRO, chap. I, pp. 134-1641. The main results concerning those operators are presented in the monograph [PIE, pp. 36-44]. I n 1967 the author has developed the theory of absolutely r-summing operators for 1 5 T < w ; cf. A. PIETSCH [ 5 ] . St about the same time B. S. XITJAGIN/A.PSLCZY~SSI [1] found the notion of a n absoliitely (r,p)-summing operator. Finally, the author defined absolutely ( r , p , q)-summing operators in the pre-version of this monograph. The famous factori7ntion theorem for ( p . q)-domi[6]. luted operators was proved by S. HWAPIEB Recommendations for further reading:
[LOT], [SEMI, rxp. 7, 27, 27bis], [SEM,, exp. 2 , 3, 16, 16b1s],[SEM,, esp. 10, 10-121. P. ASSOUAD[l], J. s. COHEN [2], CH. CCJNSTXNTIN[3], V. CRAIU/V. ISTRATESCU 111, J. DIESTEL "4, E. DUB~KSKY/M. s. RAXAXUJ.4N [I], [2], K. FLORET[I], Y. GORJ)EN/D. .J.
242
Part 4. Basic Examples of Operator ldeals
Lmrls/J. 1%.RETHERFORD [ i ] , [ 2 ] , A. GROTHENDIECK [5], G. H. H ~ R D Y /k3. J .LITTLEWOOD [I], A. JOICHI [i], &I. K ~ T O [l], 8. RwIPIE~~/A. PELCZY~~SKI [i], A1. J. KADEC[l], J. TK. LAPRESTE [l], Z. LINDEP\TSTRAUSS/L%. PF: C Z Y ~ S K I [ i ] , B. XAUREY[ 2 ] , B. MAUREY/A.PELC Z Y ~ ~ S K[I], I I~(r,9,p)Ires*
18.1.7. The preceding result leads to the following Problem. Find all exponents such that the quasi-normed opcrator ideal is regular!
[ % ( r , p , q ) , %,P,*)I
Remark. The regularity is obvious for 11 = 2; cf. 18.1.8.
18.1.8. Using the same technique as in 8.4.9 we get the Proposition 1. The quasi-normed operator ideal [%(r,z,9). S(,,,,,)]is iwjectite. The dual statement is also true.
Proposition 2. The quasi-nornzed operator ideal
[?Jl(r,p,2), S ( r , p , 2is ) ]surjectice.
18.1.9. We now prove a remarkable result which at first glance looks very surprising.
+ l/r > l/p + l/q, then the quasi-nornwrl operutor
Theorem. If 0 < 1 < cx) and 1 ideal [%(r,p,q), X(r,p,q)]is ultraatable. 00
Proof. Consider an N(,,,,,,-hounded farliily of operstcrs S , !Jl(r,p,9)(Ez, F , ) and put := sup ( N ( r , p , q ) ( , Q Li )E: I ] .Given u > q,,we choose ( I , p , 9.)-nuclearreprcsentations W
si = 2 oikark @ Y i k h=l
Such $hat 1,(ark)5 O, wq.(aik) 5 1, wP*f&&) =( 1, and IS fixed. If ok := li?n u2k,then
bil
2 C T , ~1 --.2 0, where i
U
lr(ck)5
ul 2 c2 2
2 0. (1) On the other hand, it follows from wqp(aik) 5 1 that for all finite sequences (&) with B
5 1 we have
and
248
Part 4. Basic ExRmples of Operator Ideals
This proves that Wq*(Uk)
5 1 , where
:= ( U i k ) n .
(2)
yk := ( y i k ) n .
(3)
Analogously we get
5 1, where
Wp’(&)
The proof will be complete if we can show that the ultraproduct S := (Si)uadmits the representation
z OD
S
=
akak@ gk*
1
Since 1
+ l / r > l i p + l / q we can find r, with 0
< r < ro < 00
and
l/so:= l/ro 4-l/p’
+ l/q’ > 1.
7n
By muim5 2 U i k 5 or we have k=l
Hence, given E
{5
> 0,there exists n, such that 1lro
uz}
5 E for all i E I and
n
L no.
k=n+1
Let, x = ( x J U with zi E UE, and put 7a
Uin
:=
uk(z,
a k ) yik.
I;=1
Choose bin E Ups.such tha.t [Ispi- uinli= (Sixi - uin,bjn). It follows from
that
We can find I, E U such that
Moreover,
18. Nuclear Omratom
249
Hence S adinits t.he desired representation OJ
s = 2 aka, @ ?/ka 1
Using (I), (2),and (3) we see t,hstS is an (r,p,q)-nuclear operator with Ntr,p,q)(S)5 0. 18.1.10. For every operator S E 8(El P)we put
%,p.*,(S) :=inf Wi) W q
W Wp’(Yi)
9
where the infimum is taken over all finite representations n
8 = C aiai @ yi. 1
18.1.11. Using the method of 18.1.2 we get the Proposition. N?,,,,, ii? an s-nwm on. &. 18.1.12. We now show that the s-norm N:r,p,qhis regular. Proposition. If S E &(If, P).then Nyr,p,q)(B) = N?r.p,g)(fCFS)
> 0, we consider a finite representation
Proof, Given E
n
KFS = C ciai@ yy 1
such that
I,(%)
W,,(%)
w-&3 5 (1 + E ) N:,p,*)(KFS) *
Let M be the linear span of yy, ..., ?J; E F”. By E.3.1 there exists Jo E g ( M , F ) with llJoli 5 1 + e and KFJoy” = y” whenever y” E M n H ( K F ) . If yi := J&r, then it follows from KFSX = KFJOK~BX= K F
250
Part 4. Basic Examples of Operator Ideals
that
Consequently
respectively, we have where e is a positive constant. Given S E 2(E, P)and E
> 0, we choose an ( r , p , p)-nuclear represent.ation
Q)
S = 2 o,ai @ yi 1
such that
Then there exists n wit,h
2 [(I
+ + 4 N(r,p,q)(4s. ElS
This proves that X:r,p,q)(S) (= N(r,p,q)(S). The converse estimate is trivial. 18.1.14. The following results can be checked using the same technique as in 6.8.4. Lemma 1. Let X E g(E,, F )and 8 E !R(r.p,q)(E, F ) . Then
NPr,p,q)(SX)5 X ( r . p . q ) ( @ IIXII
-
18. Nuclear Oporators
251
Lemma 2. Let S E %(r,p.q)(E, F ) and B E g ( F , Po).Then
-
2 IIBII % , p . q @ )
NP,,,,,,(BS)
18.1.15. Analogously to 10.3.1 we have the following iniprovenient of 18.1.13. Proposition. Let E' or F possesses the metric approximation property. Then
q,,p,q,(s) = N(,.,,,)(S)
for
flzz
sE W E ,F ) .
18.1.16. For spocial exponents the above result holds without any assumption on the underlying Banach spaces. Proposition 1. for all
qr.z.q)(fJ) = %,z,g)(s)
s E 5 ( E ,F )
a
Proof. We know from 18.1.8 (Proposition 1) that the quasi-nornied operator ideal
[!lt(r,2,q), N(r,2,q)] is injective. Hence, without loss of generality we may suppose that F is finite dimensional. Analogously we get Proposition 2. N:r*p,2)(s)= N,,,p>2,(s)
18.2.
for all
s E 5(E,P) *
**-NuclearOperators
18.2.1. Let 1 2 r I_ 00. An operator S E c ( E , F ) is called r-nuclear if it belongs to the normed ideal
[%,, N,1 := [%,,,I),
%,r,1)1.
Moreover, we put Nf(S) := hyr,,,lJS) for all finite operators.
18.2.2. Proposition.
[a,, N,] c [!&, P,].
Proof. Given S E Sr(E,F)and P
> 0, we choose an r-nuclear representation
co
S=
ciai@ yi 1
such that l,(ci) = 1, w,(ai) = 1, and w,.(yi) 5 (1 11~x11 2 (1
+
&)
N,(#)
K
1/7
0;
~(z,ai)lr}
+
8)
R,(S). Then
for all x E E .
This implies, by 17.3.2, that S E !&(I#, F ) and Pr(S) 5 lu,(S). The canonical map J , from C[O, 13 into L,[O, 11 is absolutely r-summing, but noncompact. Therefore 3,+ Q,.
18.2.3. As a suppleinent of the preceding statement we forinulate the Lemma. I/ s E ~ ( z Fz),, then w:(S) = P,.(S).
252
Part 4. Basic F:xsmplos of Operator Ideals
Proof. By 17.3.3 t h m arc numbers pl,...,pm2 0 such that
111
pi = 1 and 1
P u t y i := p;'"Sei. Then for any choice of
rl. ...,
)I,,,
E
X
we have
Heme w,.(?yt)Z Pr(S).It follows from m
= L1 &*ei
3 yi
1
that
B:(s) 5 lr(p:'r)W W ( C i ) wr.(yt) I P,(S). The converse inequality is evident bp 18.2.2.
18.2.4. The following inclusion is an immediate consequeiice of 18.1.5. P r o p o s i t i o n . I f rl 5 r,, t 7 w [!Jtrl, N,,] G [?RrI,XrJ. R e m a r k . Suppose t h a t rl < r2, and let (a,) E I,, \ Irl. Then the corresponding diagonal operator IS belongs to %,a(Z, Z,J. However, it follows froin S $ ?@Tl(Zm, Z,) that 8 fails to be r,-nuclear. Therefore !It,, c Rrt.
18.2.6. We now state a fundamental result which is a special case of 18.4.5. Theorem.
18.3.
[a:,-U]:
=
[p+,€',*I.
p-Compact Operators
18.3.1. Let 1 5 p 2 00. An operator S E B(E,P) is called p - m n w if it belongs to the normed ideal [Rp, K p l
:= m a ? , p . p ' ) ,
~~w,p.pf)l.
Moreover, we put KE(S) := X:m&.p,ps)(S) for all finite operators.
18.3.2. The phrase "p-compact" is derived from the following property. Theorem. A?z operator 8 E B(E,F)is p-wmpa.Ct if and only if there ex&& a mmuWive dingram
18. Nuclear Ouerators 111, this
253
case,
K p ( 4 = inf IIYII IiAll , zchere the
h / ? h 6 U m is
taken over all possible factorizations.
Proof. The necessity of the above condition follows froin 18.1.3. The sufficiency is also evident ; cf. 19.3.2 aud 19.3.7.
18.3.3. We now state a fundnrnental result which is a special case of 18.4.5. Theorem.
18.4.
[R;,KE] = [ap*, D,.].
(p, q)-Compact Operators
18.4.1. Let 1 5 p , q 2 00 and l / p 4-l / q 2 1 . An operator S f e ( E ,P) is called ( p , q)-compct if it belongs to the norined ideal [%J,q),
K ( P . d :=
where 117 := l / p operators.
+ l/q
[%r.p.q),
-
N(r,P.*)I,
1. Moreover', we put K:,,,q,(S):= X!,,p,q@) for all finite
18.4.8. The following inclusion is a special case of 18.1.5. Proposition. If p1
p2 and q1 5 qz, then
K(P,,*,)l 5 [%P*.l*)> K(P,.P*)I * 18.4.3. Leinnio. If 1 < p 5 q' < 00, then [%P1.W
K ( p , q ) ( II :; -+ )7;
5 cpaq*.
Proof. Let
be the representation described in 6.3.4. It follows from Khintxhine's inequality that
where (e) denotes the family of all e This means that wp.(e)5 2nlJ"cp,, ously we get wq,(e)5 Zn/q'cq,.Consequently
K ( p , q ) ( I2;: -+
&*. Analog-
Zi) 5 2-n211'r~-p4e) wpe(e) 5 cp,cpf.
Using the method of 6.3.5 we obtain the Proposition. If 1 < p
q'
< 00, tWen$+,,,,
R(,,,).
18.4.4. Now 18.4.2 and 18.4.3 yield the = a,,,?,. Proposition. If 1 < p , q 5 2 , thela 8(p,p)
18.4.5. Finally, we cheek estimate is required.
ZL
fundamental formula. For this purpose the following
{
0, we can find a factorization S = YSoAsuch that ljYlj l,(oi) ilAi] 5 (1 E ) N(,,l,l,(X). The diagonal operator So E B(l,, Z1) is generated by a sequence (ui) E &. Obviously we may suppose that oi 2 0. P u t S;(Ei) := (offi). h'ow the following diagram commutes :
+
s
H
+K
+
I
If l/p := l / r - 1, ehen
Si-*E Gp(Za,1,)
and S,(St-') = lT(oi)l-*.
It follows from 15.5.4 (Lemma 1) that
S;'A E G2(H,1,)
and S2(S;d2A)5 ll.(bi)7'2IIAII,
YS;, E G ~ ( zK~), and S 2 ( y S 32 ]iYil lr(oi)*'z. Hence, by 15.5.9 and lir = 112
S,(S)
+ l/p + 112, we have S E G,(H, K ) and
r S,( YS;jz)s,(S;") S 2 ( S 3 ) 5 lIYll Mod l I 4 5 (1
+
E)
N(',l,lm-
This proves that
w,T',l.l)l s [G'(H, K ) , %I.
[%,,l,l)(~,
The converse inclusion follows immediately from 15.5.2. 18.5.3. By a n obvious modifioation of the above proof we get the Theorem. On the class of Hilbert [%(?,1,2),
y r , l , z ) l = 1 6 , S'l
~ p p a c e s
for 0 < r 5 2 .
266
Part 4. Basic Examples of Operator Ideals
18.S.P. The following result is trivial. Theorem. On the du.ss of Hilbert spaces ~ ( r , z , z )= l [Gr,S,l
[%,2,2)3
for
0 I(r,q.p)l.
19.1.5. We now observe that the definition of ( r ,p , q)-integral operators, for certain exponents, yields nothing essent’iallynew. Theorem. Let 0 < r
< 00
[ 3 ( r , p , q ) . I(r,p,q)I
and 1
+ l / r > l / p + l/q. Then
= [ ! R r * p , q ) , N(r.p.q)lceU*
Proof. The assertion follows from 8.8.6 and 18.1.9. 19.1.6. The factorization theorem stated at this point is an immediate consequence of 18.1.3 and 19.1.5.
+
Theorem. Let 0 < T < m and 1 + l j r > l / p l/q. An operator S E 2 ( E , F ) i s ( r ,p , q)-integral if and only if there exists a cornmutatice diagram
.
,
Al4
so
1,
such that SoE 2(7,,, $1 i s a diagmal operator 9f the form S&i) = (oifi) with (ci) f 1,. Furthermore, A E 2 ( E . lq,) tind Y i 2(lp,F”).
I n this case, I(r,p,q)(s)
= inf ll Yii Moi) ll4l
tihere the intimum i s taken over ull possible factorizations.
264
Part 4. Basic Exemplcs of Operator Ideals
+
19.1.7. Since (r,p,q)-integral operators with 0 < r 5 00 and 1 l i r = 1/p are characterized in later section8 there only remains the following gap.
+ i/y
1. An operator S E O(E, F ) is (w,p , q)-integral Conjecture. Let, l/p -?- l / q : if and only if there exists a conimutative diagram
i
A
Y
+ such that -4 E 2 ( E , Z&) and Y E f?(Zp(I),F“) with some index set I . Hero l(p,ql denotes the embedding map froin Zq@) into I#).
In this case, b , P , , ) ( S ) = inf IIYll 1 141~ where the infiinuin is taken over all possible factorizations.
19.1.8. Now an iinproveinent of 18.1.13 is given. Proposition. Let E or P be finite dimensional. Then n?r.p,q)(S)
== I(r.p,q)N
-
for all S E W E ,P)
Proof (sketch). Tising ultraproduct techniques it can be seen froin 8.8.7 (remark) that every operator S E 9c,,p,,)(E, P)admits a factorization
.i
+ Y
such that A E 2 ( E , Eo), So E S(,,p,q)(Eo, Po), and Y E 2(Fo, F”), as well as IIYII I(r,p,q)(So) IlAIl = I,,,,,,)(S),where Eo and Fo have the metric approximation property. Indeed, we may achieve that Eo = L,,(M, p ) and Fo = L p ( N ,v ) ; special cases are treated in the followingsection and 19.1.6. Suppose that E is finite dimensional. Given E > 0, by 10.3.5 there exists B E V(F,,, Fo) with BSoA = SJ and ilBll 5 1 E. Take the factorization B = JB,, where J is the injection from M ( B ) into F,. Then it follows from llBoll = IlBll and
+
ZJ:r,,,,(BS,A) 2 %,p,q,(BoSoA)= I,r,,,,,(BoSd)
19. Integral Operators
265
that B:r,g,q)(s)
IIYII X 7 r , p , q @ J = IIYII %*p,q)(~so-4 5 lIYl1 4 r , p * q P o S o ~ ) 5 (1 -t&) IIYll I ( * * p , q , ( ~ o l )l 4 = (1 f 8 ) I(*?P&)(B)* S: , p , q ) ( K d )5
Hence K!r,p.q,(Sf 5 I(r,p,q)(S). The converse inequality is trivial. The case, where F is finite dimensional can be treated analogously or by duality. 19.1.9. The following estimates can be checked using the same technique as in 6.8.4.
Lemnia 1. Let X E $(I$,, E ) and S E 3(,,p,q)(I$, a).Then
q,p,q,(sx) d I ( , , p * q ) ( S )IlXllL e l r i n i ~2. Let S E 3i,,,p,q)(E, F ) and B E &(PiF,). Then
y,JJ,,,(~fJ) 5 IlBll I ( , , p . q ) ( S )19.1.10. The following result is an immediate consequence of the preceding lemmas and 18.1.4.
T 11 eo r e in.
13(r.p,q), r(r,p.qJ0 1% il.ill = i%.p.q)3 10, Il.li1 0 I(f,p.q)l= [%r.p,q),
N(r.p,q)19 s(r,p,q~l.
1'3.1.11. Analogously to 10.3.1 we have the following iinprovement of 18.1.15. l'roposition. Let E' or F possess the metric approximatima property. Then N:r,p,q)(S)
I(r,p,q)(s)
1
19.1.12. Finally, we state
a
for all
S E WE7 P).
result which is a consequence of 18.1.4 and 19.1.11.
Theorem. Let E' or F have the metric approximation property. Then !Rt(r,p,q) is the of B(E, F ) in 3 ( f , p , q ) ( E F ), . Furthermore, the qwwi-nomns N(r,p,q) and I ( t . p , q ) coimide a ( r , p , q ) ( EF , ).
I(r,p,q)-doSe& hull
19.2.
r-Integral Operators
19.2.1. Let 1 5 r 5 the norined ideal [3rj
00.
An operator S E B(E, F ) is called r-integral if it belongs to
I,] := [R,, Nrlmax-
19.2.2. As a special case of 19.1.2 we have the Theorem.
[n,,N,] = [S,, l,lmin.
19.2.3. The next lemma is fundamental in the sequel. Lemma. Let (Ki) be a family of compact Hausdorff spaces. Then every ultraproduct (C(Ki))ltcnlz be represented as a Bamck spuce C ( K ) with some compact Hausdorff space K .
266
Part 4. Basic Examples of Operator Ideals ~~
Proof. Since the ultraproduct (C(Ki))uis a commutative B*-algebra, the assertion follows from Gelfmd's representation theorem [DUN, p. 8761. The conclusion can also be checked by using lattice-theoretical methods ;cf. C.2.4. 19.2.4. Lemma. Let K be a compact Hausdorff space. Then, given E > 0, there exists L E g ( C ( K ) ,C ( K ) )such that Kk(L) d 1 and for i = 1 , ...,n. und
fl,
[lfi
...,f n E C ( K ) - Lfillm 5 E
..
Proof. Cover K by open subsets G,, ., G, such that l f i ( s ) - f i ( t ) l 5 E for s, t E (7,. Then there are h,, ...,h, E C ( K )satisfying the following properties (partition of the for all t f K . for all t
4 G,.
m
h&)
=1
for all t E K .
1
Furthermore, fix t , E G,, ..., t, E G, and denote the corresponding Dirac measures by dl, . ., 6,. Then
.
m
L := 2
@ 11,
1
is the required operator. Clearly ~ ~ ( 6= , )1. Moreover, it follows from
;1
1 that
1. Hence wl(hk)5 1. so me have NL(L) 5 1. On the other hand, 00
Thk completes the proof. As an immediate consequence we get the Theorem. Let K be any compact Hausdorff space. Then C ( K ) has the metric nppro;rrimation property.
19.2.5. Lemma. If X E g ( E , C ( K ) ) ,then pm(S) = l[Sll. Proof. Take a finite representation n
s=zaj@fi 1 n
such that
3 ~ ~5aIISll. i ~B y~ 19.2.4
(lemma) there exists L E g ( C ( K ) ,G ( K ) ) with
1
NL(L) 2 1 and llfi - Lfi!lm5 E for i = 1, ..., n. I-Ience
n",(s, I NO,(LS) + NL(B - LS)
+ L' llaill R
2 X o ( L ) W!I
1
llfi
- Will I (1 -L 8 ) IlfJll.
This proves that NO,(S) 5 IlSll. The converse inequdity is trivial.
19. Integral Operators
36‘i
19.2.6. We are now ready to establish the main result of this section. Theorem. Let 1 5 r < 00. An, operator S E B(E, P ) ia r-integral if and only i f me of the following statements is true: ( 1 ) There exist a compact Hausdorff space K and a probability ,it E W ( K )as well as operators A E E(E,C ( K ) )and Y E E(L,(K,, p ) , 3”’) such that
E-
W)
KFS
+P“
+
/j
,IJ,(K,p)
Jr
Here J, denotes the canonical mup from C’(1i) into L,(K, p). In this mse,
I,(@ = inf II YII llAll> where the infimum i s taken orer ull possible factorizations. ( 2 ) There e.&t a probability space (52, p ) as well as operators A md Y E B(L,(Q, p ) , B”’) such that E
KFS
i
I,
B(E,L&2,
p))
’p“
A Lm(Q, P )
f
+
I
W Q , p)
Here I , denotes the embedding map from L,(Q, p) into LJQ, p).
I n this case,
I,(fl) = inf IlPll IIAI!, where the infimum i s taken occi all possible fmtorixations. (3) There exist a meamre space (Q, p ) as well as operators A f 2(E,LT,,(O,p ) ) c u d Y E E(L,(Q, p), PI’)such that KpS‘
268
Part 4. Basic Kxamules of Ouorator Ideals
Here So i s a diagonctl operator of the form S,f In this Ccxse, I,(#) = inf
= sf
with s € L,(Q, p).
llyll llsoll Mil,
where the infimum is taken ower all possible factorizations. Proof. Let S E 3 , ( E , F ) . By 8.8.4 the operator KFS can be written as an ultraJ , where Si = Q$SJ$ with i := ( M , N ) , M < Dim ( E ) , and product N E Cod (P). According to 18.1.3 there are factorizations Si
M
1,
I.,
Sio
+
such that llAill 1, IIYill 5 1, and Nr(Sio)2 (1 c j Xr(Si). By 18.2.2 we have SioE ?&(loo, I,) and P,(Sio)5 (1 E ) I,(#). It follows from 17.2.5 that (Si0)u is absolutely r-summing and Pr((Sgo)u) 5 (1 E ) I,(S). Moreover, by 19.2.3 there exists a metric isomorphism A. from the ultrapower (l,)u onto some Banach space C ( K ) .By 17.3.5 we can find a factorization
+
+
E ) Pr((Si&) 5 (1 such that IIYoll 5 (1 presented in the form (1).
+
+
&(AS).So t,he operator KFS is re-
E ) ~
According to the diagram
we also have a factorization (2), and this again is a special case of (3). Conversely, let us suppose that KFS can be written in the form (3). Then we know from 17.3.8 that So is an absolutely r-summing operator and P,(S,) = IlSoll. Let
19. Integral Operators
269
X E $(go,E ) and B E $ ( F , Po), Since L,(Q, p) is nietrically isomorphic to some Banach space C ( K ) ,by 19.2.5 we have N&(AX) = IIAX/J.Hence, given E > 0, there X,, E O(ZZ,, L,(Q, p ) ) , exists a factorization A X = XoA, such that A, E O(Eo, and IlX,,ll llAoll 5 (1 E ) &411 IlXll. Finally7 18.2.3 yields
Zz),
+
N!(BSX)
= N:(B7K,SX)
2 x(B"PSoXo) ljAoll
IlBll IIYII P,(&O) l l ~ o l IlAOll l 5 (1 4 IPll IIYII l l ~ o l l l 4 IlXll Consequently, E 3,@, F ) and I,(&) 5 IlYIl llfloll ll4. = P,(B"YfloXo) IlAoll 2
+
Remark. The analogous characterization of co-integral operators will be given in 19.3.9. 19.2.7. Theorem. [S,, I r P = [p,,P,]. Proof. Using the fact that Fi"jhas the metric extension property, by 17.3.6 and 19.2.6, we have [!&(IT7
PJ), P,] = [ 3 , ( E ,PJ), I,].
This proves the assertion, since [Vr,P,] is injective. 19.2.8. Restating 17.3.7 we get the Theorem. [S2,12] = [Ip2,Pz]. 19.2.9. I n contrast to the preceding result, we have the Proposition. Let 1 5 T
< bo and r =# 2. Then 9,+ 3,.
Proof. The embedding map from 1, into l2 is absolutely l-summing, but not l-integral.
For the case 1 < r (remark).
< co and r $. 2 we refer to A. PELCZY~~SKI [ 5 ] ; cf. also 22.4.13
19.2.10. The following result is a consequence of 18.2.4. Proposition. If rl 2 re, then [3,,, I r J G [Sr2, ITS].
< r,. then 3,c !&
Remark. The inclusion is strict for rl 19.2.11. Proposition. If 1 5 r
< 00,
Proof.By 17.3.10and 19.2.7 ~ e h a v e 3 , G ? J 3 ~ c ? B . 19.2.12. Proposition. If 1 5 r
< 00,
then 3,c ?B.
Proof. By 17.3.12 and 19.2.7 we have 3,5 8, c Cpu. The assertion also follows from the fact that every r-integral operator obviously factors through a reflexive Banach space.
19.2.13. Finally, we state the fundamental Theorem. [3:, 1 3 = ['PI*, P,.] and [p:, P:]
= [&,
I,*].
Proof. The first equation follows from 9.1.2 (remark) and 18.2.5. The second formula is a consequence of 9.3.1.
270
Part 4. Easic Examples of Operator Ideals
19.2.14. By 19.2.8 and 19.2.13 we have the Theorem. The normed operator ideal [?&,P2]is self-adjoint. 19.2.16. Let 1 5 r 5 00. An operator S E B(E, F ) is called strongly r-integral if it admits a factorization
S
E
C(K)
Jr
+Lr(K, P )
Obviously we can also dcfine a norm for those operators. I n this way we obtain an ultrastable normed operator ideal. Every 2-integral operator is evcn strongly %integral. This is not so for r = 1 and r = 00. Moreover, we conjecture the same negative result for all r 2. Remark. Strongly l-integral operators are extensiveiy treated in [DIV, pp. 165 to 1691.
+
[a,,
19.2.16. Clearly [ST,Il]lnJis the maximal hull of NrlinJ. But it seems very Nr]lnj coincides with the minimal kernel of [Sr,IrlLnj.In other unlikely that [a,, terms, we do not know whether Nr]ln3 = [Q, 11.111 o [3,, Ir]lnJo [8,11.11]2 However, the following formula is true.
[a,,
Proposition.
[al, N,Y = [St, 11.111 o [&, I r P 0 133,11.111.
Proof. Let S E d Y J ( E F , ) . By 19.2.2 we can find a factorization JFS= Yh’J such that -4 E Q(E, E,), So E 9,(E,, F,), and Y E Q(Fo, Finj), as well as llYll Ir(So) IlAll 5 (1$- E ) rVinj(S).Let J , and J y denote the canonical injections from Eoo:= M ( A ) into E , and from F,, := J4(XoA)into Po, respectively. Then A , So, and Y induce operators A,, So,, and Posuch that the following diagram commutes:
1 \-do
Obviously A , E R(E, E,,), So, E 3 y ( E o o Po,), , and Yo E R(F,,, F ) . Moreover, IlAoll 5 IIAll, I?j(Soo) 5 IJS,), and ilY,ll 2 IlYil. This proves that [%, N,li”j G [St, /i.ll] o [S,, I,? o [R,il.i[]. The converse inclusion can be derived from 19.1.10.
27 1
19. Integral Operators
19.3.
p-Factorable Operators
19.3.1. Let 1 5 p 5 cw.An operator S E B(E, 3') is called p-factorable if it belongs to the normed ideal
[I?, L,]
:= [Rp, Kp]mrtx.
19.3.2. As a special case of 19.1.2 we have the
Theorem. [A,, H,] = [I?, LPlmin. 19.3.3. Obviously 19.1.4 yields the
T h e o r e m . [I?,, LpIdua1= [I?,,, lip,]. 19.3.4. The next lemma is fundamental in the sequel.
Lemma. Let 1 2 p < 00 and let ((sZi, pi))be n family of meusure spaces. Then every ,ultraproduct (L,(Qi,pi))u can be represented as a Bamch space Lp(Q,p) zuith some 'vzeasure q a c e (52, p). Proof. Obviously the ultraproduct (L,(Qi, pi))ubecomes a Banach lat.tice with respect t,o the natural ordering. Now the assertion follows from C.2.2, if we shoir. thnt
Ilh -txzi)uIlP = Il(Z1i)UlP
+ Il(x2d11!P whenever (xli)u
A
( x d U = 0.
For this purpose put
p. .__ .-- xli - (zli A zpi)and xg6 := zpi- (xli A x 2 i ) . Then (x:Ju have
+
= [q:li)ll and (x& x:ilip
Il(4i
+
= (xzi)wMoreover, zyi
+ Ilz;Jp.
= llz~illp
4 i ) U I P = IlC4)UliP
A
zii = 0. Consequentiy
we
So
-t lI(4i)dP.
This completes t,he proof.
19.3.5. L e m m a . Let (8,p) be n measure space uizd let 1 5 p < 00. Then, given fl, ...,f n E L,(Q, p ) and E > 0, there emkt8 L E g(Lp(Q,p), Lp(Q,p)) such that K:(L) 5 1 and l l f i - LfijIp5 F for i = 1, ...,n. Proof. Choose simple functions fy, ...,f: E Lp(Q,p) with llfi 5 ~ / 2 Then .
c!jp
me can find disjoint p-measurable subsets GI, ...,Qm such that
..
ft =
II
8&, k= 1
T-hcro
.,h , are the corresponding chara.cteristic functions. TTithout loss of generality we limy suppose that 0 < &&) < 00. Define the functions u,,..., um E p) and 91, ., 21, E Lpt(G,/l.) by Uk := p(sZk)-"' h k and vk := ,/L(Qk)-'''' h k , respectivdy. h,,
&(a,
..
Then m
IJ := 2 Vk 0U k I 111
is the required operator. It follows froin
:1
IVf'kUk
1:
m
=
2' \&IP
I?#
5 1 that
? p(Qk)-l
llhkl/: 5
*
272
Part 4. Basic Examples of Operator Ideals
Hence wp,(uk) 5 1. Analogously we have wp(vk)5 1. Consequently K;,(L) 2 1. On the other hand, Lhk = hk implies .Lfy == f 8 and therefore
As an immediate consequence we get the Theorem. Let (Q, p ) be any inearnre spuce und let 1 5 p < ce. Tlwn Lp(Q:p ) hm the metric approximation property. Remark. The result is also true for p = ca,since L,(O, ,u) can be identified with some Banach space C ( K ) ;cf. 19.2.4. 19.3.6. Using the method of 19.2.5 we can check the Lemma. If S E g ( E , Lp(s2,p)), then K#3) = IlSl!-
19.3.7. We are now ready to establish the main result of this section. Theorem. Let 1 p < 00. An operator S E &(E, F ) is p-factorable { f aad 0’)21y if there exists u commutative d i q m m
such that A E E(E, Lp(Q.p ) ) und Y E !2(Lp(f2.p),F”). Here (Q, p) 6 a suitctble measure space.
I n this case,
Lp(S) = inf IjYIi $4il, where the infimum is taken over all p o s ~ b l efacton’zrntims.
Proof. Let S E EJE, P).By 8.8.4 the operator KFS can be written as an ultrawhere Si= QCSJC with i := ($1, X ) .-M E Dim (F),and N Cod (p). product &(S~),J, According to 18.1.3 there arc factorizations
-
such that llAili 1 and ii17jl!5 (1 + E ) K,(S,). By 19.3.4 the ultrapower (lP)ucan be represented as a Banach space L,(Q, ,u), Hence KFS factors throngh A,(&?,p). Conversely, let us suppose that KFS admits the described factorization. Then by 193.6 we have K i ( B S X ) = K:(BzKFSX)5 IJB”YIJK;(AX) 5 /JBIJ IIYI;l]Ali 1IXiI f o r Y < @(E,, E ) and B E @ ( F , Fo). Consequently S E B J E , F ) arid Lp(S) 5 (/YiIIIAli.
19. Integritl Operators
273
19.3.8. Obviously, if p = 2, we obtain the ideal of Hilbert operators defined a t the point 6.6.1.
Theorem.
[Bz,L,]
= [8, HI.
19.3.9. We now deal with the ideal of co-factorable or oo-integral operators. Theorem. An operator X E B(E,F ) i s co-factorable if trrd only if one of the following statements is true: (1) There exist a eo7npact Hausdorff spnce K as well as operators A E 2 ( E , C ( K ) )crnd Y E f?(C(K),F ” ) such that
I n this case.
L,(S) = inf llYll IIAl!: where the infimum is taken over all possible factorizations. ( 2 ) There ex& a m e w r e space (a. ,u) as well a8 operators A Y E B(L,(O, p), F”) such that
< B(E, L,(O,
p)) and
I n this case,
L a m = inf IlYll
llAIl7
where the infimum is taken over*all possible factorizations.
Proof. The criterion (1) can be checked with the technique used in 193.7. Therefore it remains to show the equivalence of (1) and (2). If KFS admits a factorization (l),then
(Kp)’ Y’’KC(Ky4 = (KF,)’ K y 3 - A = KFS. Hence KFS factors through C(K)”. But, if we identify C(S)’ with some L,(B, p): then C(K)“ and L,(Q, p ) coincide. So there exists also a factorization (2). The converse implioation is evident, since every Banach space L,(O, ,it) can be represented as some C ( K ) . 19.3.10. Analogously t o 19.2.13 we have the fundamental Theorem. 1R
[a;,I$ = [ap*, ] D,*] and [%o*,,
PiPtsch. Onrratnr
D*,]= [Ep., LpJ.
274
Part 4. Basic Examples of Operator Ideals
19.3.11. Let 1 5 p < 00. An operator S E E(E, F ) is called strongly p-factorable if it admits a factorization
Obviously we can also define a norm for those operators. In this wuy we obtain an ultrastable norined operator ideal. Every 2-factorable operator is even strongly %factorable. It seeins to be likely that this is not true for p =+= 2. An operator S E B(E, P)should be called &mgly oo-factorable if it factors throzgh a Banach space C ( K ) . Furthermore, an operator AS E(E, P)is said t o be discretelyp-factorable if it adrnits a factorization
where I is some index set. The normed ideal of those operators is denoted by [&, F,]; cf. 4.7.1 and 4.6.1 for p = 1 and p = M, respectively.
19.4.
(p, q )-Factorable Operators
+
19.4.1. Let 1 5 p , q 5 00 and 1/p l / q 2 1. An operator S E E(E, F) is called ( p , 9)-factorable if it belongs to tlhe normed ideal L(P.,)I
I%Lq),
:= [%,.qb
K(p.q)lmax*
19.4.2. As a special case of 19.1.2 we have the heo or em.
r&,
K(,,,)I = [I?(,.,),
~ ( ~ , ~ d ~ ~ ~ .
19.4.3. The next result follows from 18.4.2. P r o p o s i t i o n . If pl 5 p 2 and qI 2 q2, then
-
E [ ~ ( P 2 d 7 & L(P2,Pp)1
@(Pl.T1b ~ f P , d 7 , ) 1
19.4.4. On the other hand, by 18.4.3 we have the P r o p o s i t i o n . If 1 < p 2 q’ < co,tlhen i?(2,2) f?,,,,). 19.4.5. Now 19.4.3 and 16.4.4 yield the P r o p o s i t i o n . If 1 < p , q 5 2, then I?(,,,) = R e m a r k . Let us recall that & e , z , is the ideal of Hilbert operators.
19. Integral Operators
19.4.6. Theorem. Let l / p
275
+ l / q > 1. An operator S E 2(E,P)i s (23,q)-factorabze i f
n d only i f one of the following statements i s true: ( 1 ) There exist a probability space (Q,p ) as well as operators A €
B(E,Lq>(S, p))
rrnd Y f i?(Lp(Q, p),I?") such fhat
I
A \
Here
denotes the embedding map from Lq,(Q,p ) into L,(Q, p ) .
I n this case,
L ( p , q d 4= inf II Yil l l 4 where the infimum i s taken over all possible factorizabim. ( 2 ) There exist a measure space (Q,p ) as well as operators A Y E i?(Lp(Q, p ) , F") such t h t
2(E,L,.(Q,
p ) ) and
I
.I
where the infimum is taken over all possible factorizations. ( 3 ) There exist measure spaces ( M , p ) and ( N , v) as well as suitable operators A E $(E, L,,(M, p ) ) and Y E 2(LP(N,Y), B"') such that E
K,-8
'P" A
Y
276
Part 4. Basic Examples of Operator Ideals
Here So is n positive operator frmn L,.(M: p ) into L J N , v). I n this case, Ii(P.*d@
= inf
IlYlI l l ~ o lll l 4 >
where the infimum is taken over all possible factorizations. Proof (sketch). The necessity of condition (3) can by checked by ultraproduct techniques. The crucial point is to show that every positive operator from Lqt(M,p ) into Lp(N,v ) factors through a diagonal operator. To pass from (2) to (1) we put Q0 := {co E
lslr d p
0 : s(w) $: 01 and y o ( d ):= A
[aJ 1~1'JIp]-~.
Then (Go,p,,) is the desired probability space. Finally, the sufficiency of condition (1)can be proved by a straightforward generalization of 19.2.6. For further information we refer to [SEM,, exp. 15, 161, J. T. LAPRESTI? [I], and the famous thesis of B. MAUREY [2].
19.6.
Multiplication Theorems
19.6.1. Theorem. [2(p,Q), L(p,q)]0 [@:ya',
p:ya'] 2
isp,I,].
a). Let X f &(Eo,E ) and Proof. Suppose that T E ?@'"'(E, B ) and 8 E f?(p.q)(B, C E &(G,Go).By 19.1.9 we have K&,*)(CS)5 IlClI L(P,47)(4So, given E
> 0, there exists a finit.e represent,ation
19. Integral Operators
277
19.5.2. As a special case of the preceding result we formulate the T h e o r e m . [$, Lp] o
[vy, P p ] 5 [gP, I,].
19.5.3. Passing to the injective hulls from 19.5.1 we get a multiplication formula which is analogous to 20.2.1.
Theorem. [f?&, Lf&,] o [?-@"' > Pdual q' 1C =
19.6.
Ppl.
Division Theorems
19.6.1. T h e o r e m . [ 2 ( p , qL(p,q)] ) , = [3,,I,]
0
[?@tpal,
ptyal]-l.
P r o o f . Suppose that S E 2(E,F ) belongs to SPo (g:?al)-l. Take B E g(P,Po), Lo E ~~,.,,.,(F,,Eo), and X E g(E,, E ) . Given E > 0 , by 17.4.3 we can find a factoriM ) , X o E v;Ya1(M, Eo),and zation Lo = X,Bo such that 13, E !@pt(Po,
P;~"'(xrJ) P,.(Bo) 5 (1
+
qp~,&o).
We have the diagram
and thus, by 19.1.9,
Consequently, according to 18.4.5 (lemma),
The converse inclusion is a consequence of 19.5.1.
278
Part 4. Basic Examples of Operator Ideals
19.6.2. As a special ease of the preceding result we formulate the Theorem. ,!lE
19.7.
L,] = [a,, Ipl 0
PdpUB11-.'
Notes
The concept of a 1-integral operator was introduced in the fundamental memoir [GRO, chap. I, pp. 124-1481. The theory of r-integral operators goes back to A. PERSSON/A. PIETSCE[l], and operators factoring through L, were investigated by J. LINDENSTRdUSS/A. PEECZYI~KI [l] and mainly by S. K W A P I E[S]. ~ An indispendable reference is also the famous thesis of B. MAUREY121. I n any case the reader should consult [SEN4, exp. 16, ISbi*]. Finally, the author defined (r, p, q)-integral operators in the pre-version of this monograph. Recommendations for further reading:
[LAC], [LOT], [SAF,].
D. DACIJWHA-CASTELLE/J. L. KBIVINE [l], T.GORDON/D. R. LEWIS [1], Y. GORDON/ D. R. LEWIS/J.R. RETHERFOBD [l], [Z], J. R. HOLCB[2], J. TH. LAPRESTE[l], D. R. LEWIS [Z]. K. Itbyazig~[3], A. PELCZYNSKI [5], A. RETSCH [9], J. R. RETHERFORD [2], I. I. TSEITLIN c21.
20.
Mixing Operators
In this chapter we investigate the ideal B+s,P, of (s7p)-mixingoperators the significance of which follows from
%
m(8.P)
E @P
forO= 1 and a p-normed
Theorem. operator ideal for 0
) with respect to the weak C(U,,)-topology, we have (+) for all v E W ( U F f )So . it follows froiii 16.4.3 that m(s,pl(Szi) 5 mp(zi). This completes the proof.
30.1.6. Theorem. Tlte quasi-normed operator ideal [?&,,,,, Bl,,,,)] is maximal. P r o o f . Suppose that xl, ..., xmE E and bl, ..., b, E F’. Write x:, ..., .& if zl, ...,r,,, are considered as elements of their linear span LW. Obviously we have J$xq = x8and w,(x!) = wp(xJ. On the other hand, put
N := (y E F:(y, b,)
=
- - a
..
= (y,
Then there are b!, .,b: E (FIN)’ with follows from S E aE,;(E, F ) that
bm) = 0 ) .
(9;)’b! = bk
and I,(bp) = &(bk). Now it
20.1.6. The next result is evident. Theorem. The quasi-normed operator ideal
[!Dl(s,,,),fil(8,pJ tk injective.
20.1.7. The following criterion is a generalization of the fundamental characterization given in 17.3.2. T h e o r e m . Let 0 < p 5 s < 00. A n operator S E e ( E , P) i s (s, p)-mi&?q if and there exists a constant G 2 0 such that for v E W(Up)we can fin& p E W(U,,)
only if
20. Mixing Operators
281
tohenever x E E. Here we suppose that UE#ond UFtcrre equipped with the weak E-topology and P-topology, respectively.
In thi.9 caRe,
31(s,p)(S) = inf 6. Proof. If the above condition is satisfied, then
) owP(x,).Hence ASE !Jjl(s,p!(E, F ) and lII(B,p)(S) 5 6. This means that r q s , p , ( S z i5 If s 2 1, then for every Y f W(U,.) we consider the operator J , E ?@,(P, L,(UpP,,1
*
Proof. Thc assertion follows iminediatcly from 20.1.7. 20.1.10. We now state the main result of this section. Theorem. Let l / r [?@n Pr1
S
+ ljs = l / p 5 1. Then.
[ 9 J 4 s , p ) 7 sI,s,p)l-
Proof. E’irst we claim that, given a prabability p on a compact Hausdorff space K , the canonical map Jr from C ( K )int,oL,(R, p ) is (8, p)-mixing and Mf8,p)(Jr)= 1. Let
282 fl,
Part 4. Basic Examples of Operator Ideals
...,fm
r’/s
E C ( K ) and h E L , ( K ,p) with llhllr. 5 1. It follows from p / r
+ rf/p‘= 1, and l / r + l/s + 1/21‘ = 1 that J If#’/‘
I(Jrfi,h)I
IfJ”a
+ p / s = 1,
lhlr’/”Ihr’Ip‘ d p
K
then we get
and m
m
Z I(gi, h)ls 5 J Z I f i l P K
1
lhl”
d wp(fi)P-
1
Hence m(s,p)(Jrfi)S U t i )
w,(gi)
5 wp(fi)*
This proves that J , E ?Jll(s,p)(C(K), L J K , p ) ) and M(s,p)(Jr)5 1. Finally, the assertion follows from 17.3.6, since the normed operator ideal [%ll(8,p),M,,,,,] is injective. Remark. It was proved by G. PISIER [2] that the assumption p 2 1 is essential.
20.1.11. As an immediate consequence of 16.4.5 we have the Theorem. Let l / r
+ l/s = 1/p. Then
[ m ( s , p ) , M(s,p)l
s [%r.p),
P(r.p)l.
Remark. It follows from 22.3.5 (remark) and 22.6.3 that 9Jlm(s,p) $: ?@(r,p) if 0 < p < s < 2 and l/r l/s = 1/21.
+
20.1.12. We now show that Theorem. Let 1 5 p (Q(r.p)
m(s,p) and
almost coincide.
5 so < s 5 00 and l/r + 11s = 1/p. Then
?Jll(s,,p)*
Proof. Suppose that S E ?&r,p)(E,F ) and x,, ..., x, E E. Given any v E W ( U p ) , we consider the operator J , E ?&o(F, Lso(UPr, v)) assigning to y E F the function f, with f,(b) := (y, b}. Clearly P,JJ,) = 1. Choose ql, ...,qm E L8;(UF,, v) such that l/Jvt!3x& = (JJzj, qi} and IIgills; = 1. We now defini: operators X E B(E,”.,E ) and G e(Ls0(UF7 y), ZE) by m
Eixi and
X(Ei):=
af := ((f, gi)).
1
Then iiXi1 = wp(xi)and llGll = 1. Moreover, let T E,:Z(J!
Z2) be the diagonal operator
20. Mixing Operators
generated by
(ti)
M(P,l)(T)
283
17. Applying 20.1.10 we have
5p
p m = l,.(ti)
-
Then it follows from 20.2.1 (remark) that P(Sfl,l)(f=T)5 PV,P)(S)IIXII &,l)(T) 2 P(r,p)(S) WP(Zi) Wti)
9
where 11s := l/p - l/r. By 18.2.3 and 22.6.5 we get
NZ;(SXT) =( .Wp(Zi) lP.(ti) with some constant
(r
2 0. It follows from 18.4.5 (lemma) and
that
Hence
So, by 16.4.3, we get for all Y E W(UFn). m ( s , , p ) ( w5 G W p ( & )
*
This completes the proof.
20.1.13. P r o p o s i t i o n . Suppose thut F is an intermediate space of (Fo,F,) posses-Sing J-type 0. Let 0 c p S so, s1 5 00 and 11s := (1 - @/so O/sl. If X E E(E,P A ) , then rS E ?lR(s,,p)(E, Fo) and S € !D2(sl,p)(E, PI)imply S E !?JI(s,p)(E,F ) . MoreWer,
+
M ( 8 , p ) ( 8E: -+ P) 5 M(a,,p)(S: E+
M ( s l , p ) ( E~ : + F1)'.
Proof. P u t l / r := l/p - l/s, l / r o := 1/p - l/aO, and l/rl := l/p - l/sl. Then 1/r = (1 - O)/ro O/rl. Let zl, ...,2, E E with wp(si)5 1. Given t' > 0, we have Szj = toi~,,i and Sxi = tliyli such that
+
lre(roj) lr,(tli)
+ 2 (1 +
S (1
5 1,
8)
H(s,,p)(S:E'--j, F a )
and
E)
M(sl,p,(S:E + P I )
and lvsl(yli) 5 1 -
ws,(~oi)
Furthermore, we may suppose t'hat zOi> 0, and zli > 0. P u t yfi := tTISzi. Then I,(ti)
5 l r , ( ~ o i ) l - ~L l ( t l i ) ' .
On the other hand, it follows from
I(yi, b)i that
= I(yoi, b)ll-e l(yIi, b)jo
for all 6 i F'
t i:=T
and
284
Part 4. Basic Examples of Operator Ideals
Consequently m(8,p)(szi!
2
mS(?/i)
- (1 4E ) 81(se,,)(x I : E -> $'o)'-'
~I(s1,p)(8 : E -+ P,)'.
This proves the assertion.
20.1.14. Let us define the space ideal 48.p)
:= 'pact
(m(S.P)).
20.1.15. We now prove that M(s,p) does not depend on the parameter p . Theorem.
M(s.g) for 0
:
< p < p < s < 00.
Proof. By 20.1.9 we haw M(s,p) M(s,q). Conversoly, let E E M(s,q). Then
M ( s , q ) ( I M5) M(,,,)(IE) for all M E Dim ( E ) . Noreover, 20.1.8 yields
and by 20.1.13 we get M ( q , p ) ( I I w )I )r(s,p)(I~~)'-@
nl(p,p)(Iaf)b
where 8 is determined by l/q = (1 - 0 ) l s qs.p)vMY
+ 0/p. Consequently
I %?,g)(~M) 5 M ( s , q ) ( G
Finally, it follows from the rnaximality of
[m(s,p), M,,,,)]
that E E M(s,p).
20.1.16. We now give an interesting characterization. Proposition. Let 1 < s (= 2. A Banuch space E belongs to
if and only if
B(%> E , = ' p s ' ( c O , E ) . W
Proof. If (zi) m,(E), then X(Ei):= _2: E*zi defines an operator X
B(c0, E ) . SO
1
we get a metric isomorphisin between m,(E) a i d 2(co,E ) . Moreover, using the same method as in 18.2.3 we can see that in this may ttt(8,1)(K)is mapped onto !&(~, 3). Consequently, the statements w,(E) = m(s,l,(E)and 2(co, E ) = @,.(c0, E ) are equivalent for cvery BanacIi space E. This proves thc assertion.
50.1.17. Theorrm. The operator idenl !l.Rm(s,p)i.9 proper if txnd only if s s >p.
>2
~
l
d
Proof. It will be shown in 23.3.5 that tho identity map of 72 belongs to !&2,p, with 0 < p 5 2. So !l.R(s,p) 2 !J.R(2,p, is non-proper for 0 < p 5 s 5 2. Moreover, we have ?Jll(p,p) = 0 for all p > 0. On the other hand, if s > 2 and s > p , wc Bee from 20.1.11 that !J.R(s,p) & ?&r,p) with 1/23- l / r == l/s < 112 and p 5 r < co. SO W ( s , p ) is proper by 17.9.7.
286
Part 4. Basic Examples of Operator Idcnls
20.2.3. Proposition. Let 1 5 p 5 s 5 Ern(p.,,% q p , . s , , 1 0
P S ,
00.
Then
I81 2 [3,,I,].
Proof. Suppose that T E 3 , ( E , F ) aiid S E 9.11(pr,sl~(B’, G). If X C g ( X 0 , E ) and C E g(G,Go),then it follows from 19.1.9 (lemma 1) and 20.2.2 that
N,(CSTX) 5 3X(pp,s,)(CS) R,(TX) 5 llC!j N(,~,,+S) I J T ) IlXll. Hence ST E SP(E,G ) and IP(ST)2 III(pr,,r)(S) Is(T). Remark. Using the same method we get mp;.P%
%wd 0 [%.P.,?),
for 1 2 po 5 p 5 co and l/p - l/r
I(r.p,q)l
=
E ~%o,Po,q)>
470.Po,q)l
lipo - l/ro.
20.2.4. We now state the main result of this chapter. Theorem. Let l/r
[@nPrI
+ 11s = l/p 5 1. Then
2 [ r n ( s , p ) , M(s,p)I and
“Pn PrI E [ W p c . s , ) , %pr,st)l*
Hence (l),(2), and (3) follow immediately from the preceding propositions. Suppose that T E V,(E, B’) and S E !Rr(F,G ) . Given E
F
> 0 , we take a factorization
S
+
such that I1211 &(So) /JBII5 (1 E ) N,(S). Since I , has the nietric extension property, we know from 19.2.7 that BT € 3,(E, I , ) and I,(BT) = P,(BT). Hence by (3) we get SoBT E 3,(E, I,.) and IP(SJ3T) 5 €‘,(So) IlBIl P,(T). Clearly we may achieve that Z E @(Zr, a). Then it follows from 19.1.10 that ST = ZSOBT f ?Rp(E,F ) and XP(ST)2 llZll I,(S,BT) 5 (1 + E ) Sr(8) P,(T). Finally, (5) is an immediate consequence of (4). Remark. It was shown by G . PISIER[2] tliitt (1) docs not hold in the case where l/r l/s = l/p > 1 in general.
+
287
20. Mixing Operators
20.2.6. The results of the preceding theorem are represented in the following multiplication table
20.3.
Division Theorems
20.3.1. As an improvement of 20.2.1 we have the Theorem. Let 1 5 s [ r n ( * , p ) , i%.p)l
co. Then =
[bs,
0 [@P> P P I .
..
Proof. Suppose that S E B(E,F ) belongs to ?j3T1o ?&,. Let xl, ., x , E E and b,, ., b,, E P'. Define the operator B E B(P7I:) by By := ((y, bJ). Then P,(B) s l s ( b k ) and therefore
..
d PP(B8)W p ( Z i ) d P,' Hence, according to 20.1.4, we have 8 E This proves that
[bs,P81Y 0 t @ p ,
Ppl
SR(,,,)(E,
0 P p ( 4 Wp(XI)
UbJ*
F ) and M ( s , p ) ( 8 )I p,'
0 Pp(8)-
E C f R S * P ) , %.p)l.
The conversc inclusion follows from 20.2.1. 20.3.2. The folloming formula can be checked by analogous considerations. Theorem. Let 1 5 p 5 s 5 00. Then
[m(p*,,'), N(p,.s~)= l E%p, ;v,l
0 [%a7
N P-
'30.3.3. Theoren?. Let 1 5 p 5 s 5 60. Then [m(pT,s,)7&p,,s,J
20.4.
=P p ,
Ipl 0 [a,, I,Y.
Rotos
Although the theory of ( 8 , p)-mixing operators is here preseated for the first time, most results can be iinplicitely found in the thesis of B. MAUREY[2]. See also [SEM,, exp. 13, 17, 211 and [SEX5, exp. 121. The i n c h i o n 8, c ?lRm(s,p) ~ s i t hl / r l / 8 = l / p 5 1 wm established bjA. PIETSCH [5]. The multiplication table is taken from 9.PERSSOK/A. PIETSCE [l], and the division theorems are due t o J. Punr, [l].
+
Recommendations for further reading:
-
3.MAU~XEY/G. PISIEI~ [I], G. PISIER [2], $1. P. ROSEXTII-~L [41, r71.
21.
Type and Cotype
In the following we deal with the ideals and C&p) of operators possessing (8, p)-type and (27 p)-cotype, respectively. The theory of these operators was created by B. MAUREYin. 1972. The basic formulas are !$38
!$3p f o r O < p < 8 5 2
OZ$;;
and
!& for 9 5 p < w . Moreover, we have z(s,p, = 2 for 0 < s < p < 1 which proves that the operator ideal f@, with 0 < r < 1 does not depend on the parameter r. o 'ppG
From our point of view the operator ideals investigated in this chapter are toolb to deal with absolutely summing, nuclear, and integral operators. However, operators of (s,p)-type and (2,p)-cotype also play a significant role in the structure theory of Banach spaces. On the other hand, using these concepts J. HOFFMANN-JQRQENSES and G.PISIER have characterized those Banach spaces in which the law of large numbers or the central limit theorein hold.
21.1.
Stable Laws
31.1.1. Let 0 < 8 I;2. Then by Bochner's theorem there exists a probability ys on the real line and on the complex plane such that exp (-1718) and
J exp (i.iz) d,u,(a)
= 9
exp (-Itis) = J exp (i Re (617))dp,(cu), y:
respectively; cf. [KAW, p. 3971. Let us mention that in the literature y, is frequentcly called a stable law. If .s = 2 and s = 1, then these probabilities have a Lebesgue density given by 1 exp (-lx12/4)
VG
1
and 4 1
+
241
+
(real line) IKl2)
and
1 exp (-IaI2/4)
47L
and
1
(complex plane), j42)3/2
respectively. 21.1.2. Put sf := s if 0 < s .= 2 and s+ := 00 if s = 2. Then there exist the absolute moments
21. Type and Cotype
< p < s+.
for all exponents p with 0
and
[r CSP
=
s
We have
r ~ r+)]'" ) r
289
(complex plane) ;
(9)
cf. [KAW, p. 4301. I n particular,
c2p
=;
8
[r ("-
p)]'",
-
cZ2= 2 . cZ1= 1.z
(complex plane).
31.1.3. We now prove a fundamental formula, where u ,: denotes the m-fold product of p5. L e m m a . If 0
< p < s+, then
Proof. Observe that the coordinate functionals f l , . . .. f m , defined by f t ( a ):= a, for a = ( a t ) ,are independent randoin variables on the probability space (Zm, p:) with corresponding Fourier transforms f z ( t ) = exp (- l t l s ) . Let x = ( E , ) be any m
wctor. If f := 1
tifr,then
Clearly the randoin variable g with g(cw) := cw 11x11, defined on the probability spaccb (Z, p,) has the same Fourier transform. So f and g are equidistributed. Consequently
an6
coincide. 1 g Pietech, Oprrutor
290
Part 4. Basic Examples of Operator Ideals
21.1.4. In the following we denote by ax and bx the coordinatewise product of b = ( p i ) with 5 = (li),respectively. Leainia. Let 0 < s, t 5 3 and 0 < p < min (s+,t+). Then
a, = (ai) and
cS;l
{1
1 Ilbxll,P dpT(b)}lIp
I!uxllf d,uP(a) = yc ~ l {
3-m
for all x E Zm-
xm
Proof. By the preceding lemma we have
and
Proof. By the preceding lemma we have
On the other ha.nd, it follows from 22.3.1 that
{ J l!axllf dp,”(c~.)
c ~ ’
x
m
Remark. If s = t = 2, then the above formula remains true for 0 < p f 2. In case 2 S p < 00 we get t,he converse estimate.
21.1.6. Let 0 < s 2 2 as welloas 0 < p x1, ...,xmE E , we put
< s+.
For every finite family (xi),whore
21. Type and &type
291
21.1.7. We DOW state an important result of J. H O F F N A " - J ~ R G E N S E X [l];cf. dso [SEM,, exp. 61. Proposition. Let 0 < p t(s,q)(xi)
< q < s+. Then there e&ts
a constant bspq> 0 smh tha
5 bspqt(s,p)(Xi).
21.1.8. Let 0
< p < 00. For every finite family (zi), where xl,..., xm i E , we put
21.1.9. Por completeness we mention an estimate which is analogous to 21.1.7; cf. [LIT, vol. 111 and [KAH, p. 151. Proposition. Let 0 < p < q < 00. Then there e&ts a constant bpq > 0 such that pq(xi) S bMrp(xi). 21.1.10. Pinally, a result of G . PISIER[l]is stated; cf. also [SEM,, exp. 31. Proposition. Let 1 < s 5 2 crnd 0 < p < s+. Then there exists a conatant btp > 0 such that rp(xi)5 b:pt(s,p)(xi).
21.2.
Operators of (s, p)-Type
21.2.1. Let 0 < s 5 2 and 0 < p < s+. An operator S E B(E,F ) is said to be of (s,p)-type if there exists a constant (r 2 0 such that t(,,,(S%)
5 4(%)
.
for all finite families of elements xl, .., x, E E. We put T(8,p@) := inf (r. The class of these operators is denoted by Z(s,p,. Remark. It follows froin 21.1.7 that Z(s,p)does not depend on the parameter p .
21.2.2. Using standard techniques we get the Theorem. [ Z ( s , p ) , T ( s , pis ) ] a normed operator ideal for p 2 1 and a p-normed operator ideal for 0 < p < 1. 21.2.3. Theorem. The quasi-normed operator ideal [ Z ( s , p )T,,,,,] , is mciximal. Proof. Let S E zr$(E,F ) . Observe that Cod ( P ) , the set of finite codinicnsional subspaces of F , is directed. We have
...,x,
for al,..., amE Y and xl, xl, .., x,. Then
.
1 I)*
E E. Let iM E Dim (E) be the linear span of
292
Part 4. Basic Examples of Operator Ideals
So, by Fatou’s theorem, we get
3 )and ‘J!(8,p)(4 5 Tr$(fi). Hence S € Z(s,p)(E,
21.2.4. Theorem. The quasi-nomzed operator ideal [Z(s,p,,T(,,,,] i s injective and surjective. Proof. The injectivity is evident. To check the surjectivity let S E Z;FA,(S, F ) . Given xl,..., x, E E and E > 0, we choose xy, ..., :x E ESurwith xi = Qpz! and ll4ll I (1 4 IIXill. Then
+
t(s,p)(&)
= t(s.p)(sQ,&)
I T(s,p)(fiQ~) l a ( 8 ) 5 (1
+
E)
TSti,(S)&(xi) *
Hence S E Z(,,,)(E,P)and T(s,p)(S) 5 TYZ’,,(S). This proves the assertion.
21.2.6. We now state the basic result of this section. Theorem. Proof. If xl,
Tt$;l G [!)J M(s,p)l. l(,,,), ...,x, E E and bl, ...,b, E F‘, then by 21.1.3 we have
= t(S,P)(fi’bk) l q x i )
5 T(S,p)(s’)1s(bk)W p ( X i ) ‘
Hence the conclusion follows from criterion 20.1.4.
21.2.6. Proposition. Let 0 < p < so < 5 5 2. An g e r d o r S E B(E,F ) is of (so, p)-type if there exists a constant cr 2 0 such that
(*I
r,(Szi) d 4(4 for all finite families of elements xl, ...,x,,,E E . I n this case
T(s,,p)(45 css0c;; P r o o f . By (*) we have
inf 6.
21. Type and Cotype
Observe that ,uE is invtiriant under all transformations follows from 31.1.5 that,
2 GS,C;-d IS&!)
a
--f
ecc
293
with e € 8". So it
*
'l'his conipletes the proof. Rcinark. Operators satisfying (*) with 0 < s bc of Raileinacher (s, p)-Lype; cf. G. PISIER [l].
52
and 0
p21.
31.4.6. Now it follows a famous Theorem. Let 1
5 p < 00. I’hen
[ a ( z . p ) >C(2,P)I 0 [%.P)>
T(2,P)I
s [f?2,
1421.
Proof. By 21.3.5 and 21.4.5 we have
C(,,,)]
0 [ 2 ( 2 . P ) , T ( 2 , P ) I 0 [V2,
s
P21dua1
EV2, p21.
So 19.6.2 yields the assertion. 31.4.7. As a n imniediate consequence of 21.3.4 and 21.4.5 we get the main result of this sectioii. Theorem. Let 2 s p E6(2,P)> C ( 2 , P ) I 0
< m. Then
[PP, PPI s
p21.
[P2?
21.4.8. Next the adjoint version of the preceding forniuln is given. Theorem. Let 1 < r, =( 2 . Tlwu [ 3 2 , 1 2 1 0 W(2.pf)’ C(2,P*)l
EP
p ,
&I
*
21. Type and Cot.ype
297
Proof. Suppose that T E C(z,p)(E, F ) and S E &(F, a).Moreover, let C E g(G, Go), Lo E !&(Go, Eo),and X E S(Eo,E). Then, by 18.4.5 (lemma) and 19.1.9, we h a w
!trace (8TXLoC)j = /trace (TXLoCS)I5 P2(TXL0)M;(CX) -
C(2,p)(T) llxll BP*(Lo)llcil I
2 W .
ST E Sp(E,G ) and I,(ST) 5 12(S)C,,,,*)(l').
Hence, it follows from 19.2.13 that
31.4.9. We now state an immediate consequence of 20.3.1 and 21.4.8 as well as of 19.2.7 and 19.2.8. Theorcin. If 1 < p 5 2 , then W(2,P*),
C(,>,*)I c=
[?JJ$Z,P),
M(2,p)l.
does not depend on p. So we have indeed Itemark. Let us recall that C(z,p) %l&z,q)for 0 < p < 00 and 1 < q 5 2. However, it seeins to be unknown whether this inclusion also remains true for 0 < q 5 1.
21.6.10. Pinally, we establish a supplement of 21.2.5 and the preceding result. Proposition. If 1 < p
< 00, then ; ; :% :
6(2,p*,. Proof. Suppose that S c Z$'$i(E, li') and xl,..., x , E E . Choose b,, ..., b,,' t 3'' such that (&xi, bi) = llSxi112 and l]biil = [j8zill. Since the coordinate functionals f l with f i ( a ):= mi form an orthogonal family in L z ( X m,up), , we have tn
I~(SZ~)' = (SZ~,b,) 1
= CG'
j"
(2
1
m
2' akS'bk
& t ~ i ,
1
, x m
I C2,%,*C&,,*)(Xi)
dpy(cx)
t ( 2 , P W k )
- C 2 p % p * C 2 q , , p ) ( ~ ' ) I,(bk)
4 2 , P * W
*
Therefore, since 12(bi)= l2(5z,),it follows that 12(8Zi)
1dua.l r c z p c z pr;c;* __ q2,pJs) t(Z.P.)(4
*
Hciicu X E a ( 2 , p * ) ( F E ), .
Remark. We will prove in22.3.4 that theidentity map of El is of (2,p*)-cotype. =I=6(2,p*). Therefore On the other hand, this map cannot belong to
21.4.11. A Banach space is said to be of (s,p)-cot?ype if it belongs to the space ideal C(8.P)
:= Space
(a,,,,)) *
Remark. Let us recall that the parameter p is indeed superfluous; of. 21.4.1. (remark).
298
Part 4. Basic Examples of Operator Ideals
21.5.
Notes
From the very beginning the scqnence of Rademacher functions and Khintchine’s inequality have played an important role in functional analysis. It was the idea of J. BRETAGNOILE/ 1). DACUNHA-CASTELLE/J. L. KRIVINEEl] to extend this machinery by using other sequences of independent random variables. The concepts of Banach spaces with super- and subquadratic averages have been introduced in a paper of E. DUBINSKY/A. PEE,CZYI$SKI/H. P. ROSENTHAL [l].The final definition of operators having (8.p)-type and (s,p)-cotype was made by B. MAUREY [2]. Important contributions are also due t o T. FIGIEL/G.FISIEE[i], B. MAUREY/G.PISIER [I], 121, and G . PISIER [I], [3], as well as to J. HOFFMANX-JBRGENSEN [I] and s. K W A P I E [71. ~~ Recommendations for further reading:
[KtzC], [KAH], [SEM,, Denx journhes p-radonifiantes], [SEM,, exp. 5-7, 8, 15, 17, 221, [SEM,, exp. 3, 6, 8, 24, 251, [SEM,, exp. 3, 4, 6, 71, [LIT, vol. 111. A. BADRIKIAN [I], s. CHEVET [2], J. HOFFMANN-J0RGENSEN/G.PISTER [I], M. J. KADEC [I], M. J. KADEC/A.PEI,GZY&SKI [i], W. LINDEIA.I’IETSCR [i], J. LINDENSTRAUSS [l], B. MAUREY [I], E. BI. NIAISHIN [l], A. PELCZYT~SKI [6], J. PUIIL[l], J. R. RETHERFORD [el, H. P. ROSENTEAL
[a], [7].
22.
Operators in L,-Spaces
Given an operator ideal on the class of Banach spaces we niny ask whether its Hilbert space part coincides with some operator ideal 6,.The answer is affirmative for most of the known ideals of absolutely summing, nuclear, and factorable operators. However, there is also a counterexample due to G. BENNETT. Next we introduce so-called L,-spaces which are closely related to those defined by J. LINDENSTRAUSS and A. PELCZY&SKI. The main purpose of this chapter is t o investigate diagonal operators from I, into 1,. Using the concept of a limit order we give almost necessary and sufficient conditions for those operators to be absolutely r-summing, r-nuclear, p-factorable etc. According t o an important theorem of H. KONIGthese results can be carried over to weakly singular integral operators.
22.1.
Operators in Hilbert Spaces
22.1.1. It can easily be seen that, for 0 < p absolute moments
< 00, there exist the n-dimensional
We have
and
Observe that c y . = c Z p ;ef. 23.1.2. P u t a,, := c!$/cg). Then 1
alp 2 a2p2
1
... 2 0
and lim n1/2anp = c+&'. It
22.1.2, Let 02" be the normalized rotation-invariant measure on the n-dimensional unit sphere S:. L e m m a . If 0 < p
< co,liken
800
Part 4. l3i~sicExamples of Operator ldcds
Proof. Thc above forinula can be checked by using polar coordinates. It also follows from 21.1.3 and
22.1.3. Lemma. If 0 < p
< 00, then. P p ( k Zg ->
Z;)
=z
a;;,'.
Proof. Using 17.3.2 and the preceding lemma we get, P J I : 1% -+ 1;) 5 Q;;,'. To establish the converse inequa1it.y we suppose that
where p is some probability on the closed unit ball
\ Ilxii:
1=
CZG;(~)
U;.Then
5 op j J ~ ( xa>lp , da;(z) d,u(n) u; s;
S;
J iiailg d,u(u) 5 o p n ~ ~ .
GPUE~
U;
so
2 c and by 17.3.2 we have P,(I : 1;
'19.1.4. Lemma. I f 1
D ( p , q ) ( IIf:
-+
)!Z 2 a;;.
< p 5 q' < 00, the%
-> )Z;
= 0;;~;;
and
L(p*,q,)(I: 1; 3 Z;) = ru;Ctbptinq.
Proof. By 17.4.3 we have
5 P#: z;
D ( p , q ) ( I2;: + )z; If the operator A,, E i?(Z!, then llAlapii= anP.Let f l .
L,(S;,
--f
If) P y y I : z'; -+ )z; 5 a;;.;;,'.
o;)) is defined by
=
C ei @ f i . 1
Since ( f i , f k ) = n-lcik, we get the &gram
Hence
Anpz:= f,, whero /$(a):= (2, a),
..., f, denote the coordinate funotionals fi(cr)
n
An,
(1) := ai. Clearly
22. Operators in Lp-Spaces
301
Finally, it follows from n = trace ( I ) 5 D ( p , q ) ( I I!:
3
I;) Ii(P.,qt)(I: 1; -+ Z;) 5 n
that we have equality in (1) and (2). 32.1.6. Lemma. If 1
Lf,"b,(l:z;
3
p
2;)
q'
< 00, then
= unq,u;I;:.
Proof. Let the operators Aw9and A,, be defined as in22.1.4. Then we have t'he diagram T
Since .;;Anp is a metric injection, it follows that
;z
g (p,q)(I: nj -+2;)
(*I
5 unq*a;;pl*
By 19.5.3 we have ;;ZC
= P,(I :
;z
3
z;) 5 L$,\)(Z :z;
--f
2;) P y ( I :z;
3
2;) 2 a;;.
Therefore equality holds in (*). 22.1.6. Lemma. If 0 < I, < s 5 2, then
M ( 8 s P ) (2;I : 3 Zg)
= T ( s , p ) ( IZt: ->
)2;
= u,~u;;,'.
Proof. Let xl,...,x, E ;2 and write xi = (fil,
...,E i n ) . Then
302
Part 4. Basic Examples of Operator Ideals
Hence, by 21.2.5, we have
B€(8,p)(I: 12" + 1;) 5 T::$(I: Z: + Z;)
= T ( 8 , p ) ( I1;: + )Z:
5 (x,,u;;P~.
(*I
On the other hand, 20.2.1 implies
P J I : 13" -+ ZP) 5 3 q P ) ( I I;: -+ 1;) P8(I:22" -+ z;). Therefore u;;
5 i?I(s,p)(I: 1; 3 )z; a;;.
So equality holds in (*). Remark. It turns out that TC2,JI:I:
22.1.7. Lemma. If 2 2 p
p ) defined on CI suitable measure space (Q, p). Proof. We see from the preceding theorem that all L,-spaces with 1 are reflexive.
< p < ca
22.2.4. Clearly 19.3.3 yields the T h e o r e m . (Lp)dua’ = L,,,
32.2.5. The next result is a consequence of 19.4.4. Theorem. If 1
< p < m, then L, C= Lp.
22.2.6. The following theorem implies that the Banach space ideals L, and q, p $: 2, and q 2. non-comparable for p
+
+
T h e o r e m . L, n L, = F for 1 5 p
L, are
< 00,
F for 1 < q 5 cw,
L, n L,
=
L,nL,
=L,
for l < p , q < O O , p + q g .
Proo-f. Let E’ E L, n L,. Then 22.4.2 yields 2 ( E , E ) = !&JEl, E ) for 1 5 p 5 2 and B(E, E ) = ?&(E, E ) for 2 < p < r < 00. Since all operator ideals ?&are proper. we see that E is finite dimensional. Hence L, n ,L = F. The formula L, n L, = F with 1 < q assertion is proved in [LIN, p. 2011.
22.3.
5 00 follows by duality. The remaining
Banach Spaces of (s, p)-Type and (2, p)-Cotypo
23.3.1. We begin wit.h a n elementary L e m m a . Let ( i l f , , ~a)d (N,v) be a n y measure spaces. Suppose that the scnlnr function f is ( p x v)-?tzeasurableon ( M x N ) . If 0 < p < q < 00, then
P r o o f . Assume that the right-hand integral is finite. Then by f ( a ) := If(&, .)I” we define a function f on M taking values in L,,,(N, v) almost everywhere. Now the required inequality follows from
and
22. Operators in Lp-Spoces
32.3.2. P r o p o s i t i o n . If 1 5 q 5 2 and 0
4E
305
< p < s < q, then
T(8.P).
Proof. Let ( 0 , ~be) any measure space. If fi, , . . : f m E L,(Q,p), then by E.5.2 we have
Hence
This means that i o q ( f i ) 5 l&fi). Since rp(fi)5 rq(fi)it follows from 21.2.6 that Lq(Q,p) has (8,p)-type. Moreover, '(S,P)(ILq(Q,p)) < =c
(*)
qSc-1. qp
22.3.3. Proposition. If 2 5 q
< 00
and 0
< p < 00,
then
L, E T ( 2 , P ) . Proof. Let (Q, p ) be any measure space. If f l , ...,f m E Lq(Q,p), then by 21.1.3 and 22.3.1 we get
Hence bq(Q, p ) has (2,q)-type and therefore (2,p)-type with 0 < p
< P 5 P. If 1 5 q 5 2 a d 0 < p
(*I
T(2,P,(Lq(Q,,,) 5 C2*Gpl for 0
22.3.4. Proposition.
< m. Moreover,
co,then
4 E C~2.P). Proof. Let (Q, p ) be any measure space. If f l , and 22.3.1 we get
20 Pietach. Operator
..., f,,,
E Lo@, p ) , then by 21.1.3
306
Part 4. Basic Examples of Operator Ideals
Hence LJQ, p) has (Z,q)-cotype and therefore (2, p)-cotype with 0 Moreover,
< p < co.
s
C~Z.P)(IL,(S),,)) e2pcG1 for !2IP < 22.3.5. Summarizing the-preceding results we have the fundamental T h e o r em. Lq E
M(2,p)
for 1
(*)
5 q 5 2 , 0 < < 2,
Lq E M ( 8 . p ) for 2 S q
< s',
0 21, v)
Pinally, t,he assertion follows from (*!
22. Operators in L,-Spaces
307
I n order to verify this estimate we put 8 := sols. By 20.1.8 and 20.1.13 we get
M ( s , s o ) ( I2;: +-):Z 5 X(m,80)(I: Zz -+ Z;)'-'
5 Pso(l:z;
M(so,so)(I: Z; +-
2 nl-5
3 2;)l-e
and
B€(8*p)(I: 2; --f 13) 5 ilx(80.P)(I: z; -+ I;) M(s,so)(f:z;
-+ I:).
Consequently
M(so,p)(I: 3 );z
5
implies = < ,) p,tn+(s-so)iso. &,p)(I: -+I
So (*) follows by applying 14.4.3. R e m a r k . The limit order of
22.4.
will be investigated in Section 22.6.
Absolutely +Summing and ?--IntegralOperators
22.4.1. It has been proved in 17.3.9 that the operator ideals ?&strictly increase with the parameter r 2 1. For special Banach spaces E and F , however, the components &(E, F ) are constant on certain intervals. Theorem. Let It and P be arbitrary B a m c h qaces. Then
5 u 5 2,
0 < r 5 2,
!&(Lt',F ) = ?&(L,, 3 ) for
3
!&(L,, F ) =!&(La, a)
for
2 5 u 2 00, 0
v 2 ( E ,L,) = v r ( E ,L,)
for
15v
5 2,
< r < s < u', 2 5 r < 03'.
P r o o f . The assertion follows from 20.2.1 and 22.3.5 as well as from 21.4.7 and 22.3.4. Remark. If 2 < v 5 00, then there cannot exist any interval A such that V r ( E ,L,) is constant for r E A and all Banach spaces E ; cf. 22.4.12 and 22.4.13,
22.4.2. We now state the adjoint result. T h e o r e m . Let E and F be arbitrary Banach spaces. Thew
3 , ( E , L,) = 3 , ( E , Lo) for
15 v
5 2,
3 w ( E 7 L w ) = 3 r ( E , L vfor ) 2 5 ~ 5 0 9 ,v < r s 0 3 )
S2(L,,F ) = 3,(L,, P) for
1 5 ' 1 ~5 2 ,
1 < r 5 2.
Proof. The assertion follows from 20.2.3 l n d 22.3.5 as well as froin 21.4.8 and 22.3.4. R e m a r k . I n particular, we have i?(Lm)A,) = p2(L,, Lo) for 1 5 ZI 5 2 and C(L,, L,) = !&(L,, L,) for 2 < v < r < 0 0 ; cf. 20.1.16. 20"
308
Part 4. Basic Examples of Operator Ideals
22.4.3. The next theorem is a n immediate consequence of 22.4.1 and 22.4.2. Theorem. Let 1 5 u, v 5 2. Then pr(Lu,L,)with 0 < r with 1 < r 5 co do not depend on the parameter r.
< 00
and 3,(Lu, L,)
22.4.4. We now prove the famous Grothendieck T h e o r e m . O(L,,L,)= 'pr(L,, L,) for 0 < r < m. Proof. By 22.4.1 we have p,(L,, L,) = !&(I,,, L2) for 0 < r 5 2. so it remains to check B(L,, L,) = @,(L1, h,).Let S E B(L,, L,) and (xn)E m,(Ll). Define the operator X E B(&, L,) by
L,) Since the dual of L, is a n Lar-space, it follows froin 22.4.2 that X E (Ippl(Z,, and therefore S X E Ip:m1(Z2, L,). Hence, by 19.5.2, we get S X E !&(&, L,). This proves that (Sx,)= (&'Ken)is absolutely 2-surnmable. So S E !&(Ll, L,). R e m a r k . The first proof of this theorem understandable for average mathematicians was given by J. LINDENSTRAUSS/A. P E L C Z Y [l]. ~ KSee I also [LIT, vol. 1. p. 681.
22.4.6. The so-called Grothendieck constant cG is defined to be the infimum of all numbers e 2 0 such that Pl(S) e /lSl\whenever S E I!@, Z2). Although nobody needs the exact value of Grothendieck's constant, everybody likes to know it. Our method yields cG 5 M[,,,)(I: 1, + 2), lb&,11(1: 2, + 21). R e m a r k . I n the following we list some upper bounds of Grothendieck's constant which have been obtained until now :
A. GROTRENDIECK [4]
< 2,302 A. GROTHENDIECK [4] < 4,604
R. E. RIETZ[ 11 < 2,261
J. L. KRIVINE[2] < 1,782
(real case)
S. KAIJSFJ~ [l] 0, then
n 5 A*(I: 1;
--f
l:) A(I: 1; -+ 1;) 5 e*~nl(l['.u,v)+&enl(~,v,~)+"
where e and e* are some constants. Remark. For most of the examples identity holds in the above inequality. HOWever, there are normed operator ideals [%, A] for which A(%*, u,v) A(%, v, u) = 2 ; cf. H. MONIG [2].
+
32.4.11. Lemma. Let 1 5 s
PP(k1,". -+
z); g
< 2 and 0 < p < 8. If E > 0. f l w i
Q?zl/s+&,
where e is sorne constant. Proof. Define a,, by l/so:= l/s + E . Clearly p .< so < s, if Then 22.3.6 implies P,(I : 1; -+ )z;
5
,,)(I: ,;z
--f
z;,)
P s 0 ( l ,:;z ->
E
is sufficiently small.
);z 5 c,,oc,-aP,o(I:79 + 1:).
22. Operators in Lp-Spacee
Now the assertion follows from
cf. [SEK, exp. 311. Protract,ed computations yield
P,(I :;z -+ 2,")
",(n log n)"S;
G . BAUMB.4CH/W. IAIXDE [I]. 22.4.12. P r o p o s i t i o n . If 1 5 r 5 2 and 2 5 u 5 00, then Cf.
15 v
5 u', u'2 v 5 0 0 , 5 u 5 co, 1 2 v 5 r ,
for 2 5 u 5 r ' , l/u' for 2 5 u 5 r ' ,
1jv
4cpn u , v) =
l/v
for
7'
Moreover,
A(!&,
u,?I)
+ A(%,,
u, u ) = 1.
Proof. It follows from 14.4.6 and 22.4.11 that
+
A(?&,
u,v) 5 I,(!&, u, u') I@, u',v) 5 l/u'
A(?&,
u,w ) 5 A(&
and u, u')
+ A@*,
for u' 5 v
v', v) 5 1/v for v I ,u'.
On the other hand, according t o 18.2.2 and 22.4.7,
A(@,,
u,V ) 5 L(Bj U, 00)
Y W T ,
u, v) 5
+
jb(@r,
60,
r)
+ A(E>r , v)
yields
l / v for r 2 v , l / r for r
5 v.
Consequently
4%
u,a ) 5
I
l/v for 2 5 u 5 Y', 1sv 5 u', l/u' for 2 5 u 5 r ' , u I 5 v 5 m ,
i
f o r r ' ~ u ~ c le s, z i g r . l/r
for r ' g u s c o , r s v s w .
Analogously it follows that
l(%n,*, b, U ) 5
1lu l/r'
1 u, for r' 5 w.
for r'
Moreover, &(I: Z; + Zi) = 1 and
A(%,,
u,
u)5 n(c,v, 1)
+ A(%n,,
1, 2)
Q
+ A(2,2 , u)
311
312
Part 4. Basic Examples of Operator Ideals
imply
A(?&,,
v, u)5 l/v‘
for 2 S u 5
00.
Summarizing the above estimates we have
A(!%*,
0, u)5
l/v’ €or 2 5 u 5 r ’ , 1 5 v 5 u‘, l/u €or 2 s u g r ‘ , u ’ ~ v ~ m , l/v’ for r ’ s u 5 o c , , l s v s r ,
I I
llr’
for r ’ s u 5 o a , r ~ v ~ o c , .
Applying 22.4.10 we see that identity holds in (1)and (2). Hence
A(!@,, u,v)
+ A(%,.,
v, u ) = 1.
The remaining case, where 1 5 u 5 2, can be treated by 22.4.1 and 22.4.2. Then we get I.(!@,, u, v) = A(!&, u, v) and A(?&, v, u)= A(%*, v, u). This yields t,he diagrams :
1
f
22.4.13. Proposition. If 2 < r
!
< oc, and 2
l/u‘ for 1 S u s r ‘ ,
3.(!@,, u, 4 =
where
e
:= 1/‘
e
A(!&, u,v)
5 m, then 2 ~ ~ 5 0 0 ,
l/v
forr‘sug2, 2 5 v 5 r r , for 2 S u S m , 2 5 ~ 2 1 ,
Ijr
for r’ 5 u
I
00,
l/u) ( l / v - l/r) + (l/r’ -112 - 1/r
Moreover,
v
+ A(%r,,
21,
u)= 1.
r
S v 5 cyi,
22. operators in Lp-Spaces
313
Proof. First of all we mention that the most complicated case, where r' < u < 2 and 2 < v < r, has been recently treated by E. D. GLUSKIN[2]. See also E. D. GLUSKIN/A. PIETSCH/J. PUHL [l], and A. PIETSCH [23]. In the following we only deal with the remaining values of u and v. Since we have 5 A(%,, u,w) the estimates from above are immediate consequences of 22.4.12. Hence l/u' for 1 5 u 2 r', 2 5 v 5 00,
A(!&u,v)
i
W & , U , VIjv ) ~for l/r
2 5 ~ 5 0 0 2, s v S r , forr'susce,rsvsco.
(1)
As in the preceding proof we get
CLld
A(%,.,
v, u) 5
l/v'
for 2 5 u 5
l/u
for l s u s r ' ,
00.
Consequently 2svVco,
l/w' for 2 5 u 5 0 0 , 2 5 w 5 Y , l/r' for r ' ~ u s c or,s v o m . According to 22.4.10 identity holds in (1) and (2). Hence
+
I.(?&, u,v) v, u)= 1. The remaining case, where 1 5 v 5 2, can be treated by 22.4.1 and 22.4.2. Then w, u ) = v, a). we get A(?&, u,v) = A(?&, u,w) and This yields the diagrams: Ib(%,,
r
r' I
3 14
Part 4. Basic Examples of Operator Ideals
Here Q' := 1/p -
(1/u - l/r') (1/ r - l/v) l/r - 112
R e m a r k . The level curves in Gluskin's squares are hyperbolas. R e m a r k . Since I.(?&, u , v) < A(&, u, v) whenever r' < u < 2 and 2 < v < P, we see that ?&=# Srfor 2 < r < 00, and, of course, also for 1 < r < 2 ; cf. 19.2.9.
22.5.
p-Dominated and p-Factorable Operators
22.5.1. Theorem. If 1 5 u 2 p 2 v 2
00,
then B(L,, L,) = B,(L,, L,).
Proof. Let S E B(L,,, L,) and 1 5 p 5 2. If X from 22.4.1 t,hat X E VY'(E, L,,).Now 19.5.2 yields
W ( E ,La), then it follows
HX E @Pal(E, L,) 5 &(E,L,) G 3 J E , 4). Using the formula Bp = S p o (@y we )get 1S E B,(L,, L,). The case 2 2 p 2 00 can be treated by passing to the dual operators. R e m a r k . The above result means that
P r o o f . The estimates from above are consequences of
On the other hand,
we get the estimates from below. Now by 9.1.8 and the preceding lemma we have the Proposition. l/v-l!2 for 1 5 ~ 2 2 ,1 5 ~ 2 2 : for I S u U 2 , ~ ~ v ~ c A(i?2, u,a) = l l v - l l u for 2 5 ~ 5 0 0 1, 5 ~ 5 2 , ( 1 / 2 - 1 / u f 0 r 2 ~ u ~ 02 05 v, I C w ; .
c
,
22. Operators in Lp-Speces
315
Moreover, I.(&, u, v)
+ qa,. v, u)= 1 .
u
i
///'/I
v
1 2
A(&, u, v)
0
for 1 5 ~ 5 2 ,
112) - 11.
for 2 5 u 5 p ,
=, d
for 2 g u s p , for Z S u U p ,
0 l/v-l/u
i
where 0-:=
f o r p s ; ' ~ 5 m 1. 5 1 ) S p .
l , p - l i u for p 5 u 5
00,
p 5 z) 5
L i .
(112 - 1/20 ( 1 / u - l / p ) 1j2 - l i p
Mort over,
qep,u,v) + f ( P p ' 11,, u ) = 1. Proof. First of all we mention that the most complicated case, where 2 < u < Y aiid 2 < v < p , has been recently treated by E. D. GLUSKIN/A.PIETSCH~J. PUHL [I]In the following we only deal with the remaining values of 91 and T. The estimates from above follow from
A(!&,
91.
v) 5 A&?,,
and l.(i?,,u,23)
u,c)
On the other hand, !&,, o 2, o
5 ?.(a,z i , p ) f i ( 2 . p .c ) .
Qpl & 3 implies e
A ( ! p y , q, u)+ Ace,. u.
27)
+ A(@,,
Using the known limit orders of @ !,, estimates from below.
and
0,
p)
2 1.
vy,by
it
suitable choice of q . w e get the
316
Part 4. Basic Examples of Operator Ideals
This yields the diagrams
? Here g'
:= 1 -
(l/u
- l/p') (1/2 - l/v) V P - 112
22.5.4. The above result can be carried over to the case where 1 < p j ( eP'I U,P ' )
=:
2(EP, of, u') and A&*,
< 2 by
a,v) = A D p , v', w').
Since 2 , = 3- and 3, = PI,we also know the limit orders of 2, and of 2, and 3,. 82.6.
as well as
Absolutely (P, p)-Summing Operators
22.6.1. We now prove ib classical result of W. ORLICZ El]. Proposition. If 1 5 q 5 2, then L? E P(2,ii. Proof. By 20.1.11 and 22.3.5 we have L, E M(z,l)2 P(p,l). However, there is also a direct proof. Let (Q, p ) be any measure space. If fl,
So LJO, p ) has the Orlicz property.
...,f,,, E L,(Q, p), then
22. Operators in L,-Spaces
22.6.2. P r o p o s i t i o n . If 2 5 q
317
< 00, then Lq E P(q,l).
P r o o f . Let, (Q, p ) be any measure space. If f l ,
...,f m E L,(Q, p), then
So L,(O, p) belongs to P(,,l). 22.6.3. As a counterpart of 22.3.5 we now have the Theorem.
Lq E P(r,p) for 1 5 q 5 2 . 0 < p < 2 , l/p - l/r = 1/2, L, P ( r , p , for 2 2 q < w . 0
where g,,, is sonze constant.
Proof. Put. q := up’, s := ur‘ and so := ur& where 0 < u < 2/p’. Obviously 0 < u < so < s < q < 2. If b,, .,a, E F’, using 21.1.5 and the preceding sublemma we have
..
So the sublemma yields
22. Operators in L,-Spaces
319
In the real case we get
The same limit appears in the complex case. 12.6.3. Theorem. If 1 5 r
< ro < m, then p(r,l)(Lm, F ) Qr0(Lm, F). such that 1 < p1 < rl < ro and l/pl - l/rl = 1 - 1.
Proof. Choose p , and r, Then & p(rl,pl). On the other hand, the preceding lemma as well as 17.1.3 and 19.2.5 imply
v(r,l)
%l,p,)(L
F ) G V r o ( LF >)
Remark. The above theorem means that ?&r,l) o 2, E
32.6.6. Proposition. If 2 5 r
4 q 3 ( r , 1 ) >u,21)
=
< bo, then
lo
I/w-l/u for l z u z r , 1 $ w 5 u , for 1 5 u u r , u 4 v s m , llv - l/u for r 2 u 5 0 0 ; 1 5 v 5 r , l\r-l/u
for
r S u Z m ,rSvSca.
Proof. By 22.6.1 and 22.6.2 we have
A(?&r,l), q, a) = 0 for 1 5 Q 2 r . Now it follows from 4@(T,l),
+ WL u>v)
u,v) I 4V(T,,,,u,u)
that
This proves the estimate from above.
:{20
Part 4. Basic Examples of Operator Ideals
By I,(ei) 5 P(rrl)(I: -+ I:) wl(ei), we have nl/r5 P(r,l)(I:1; -+ ZEj n'!U. Consequently I(!&r,l), u,v) 2 1/r - l/u. Moreover, u,v) 2 3.(8,u,v). This yields the desired estimates from below. We now give the corresponding diagram :
29.6.7. The following result is due to G. BENNETT [l]and B. CARL[l].
+ l / u - l/v.
Proposition. Let 1 5 u 5 v 5 2 and l / r = 112 I from 1, into I, is absolutely (r, 1)-summing.
Then the embed-
ding map
Remark. The spccial cases u = 1, u 6.5.4 and 22.6.1, respectively.
=
2 and 1 5 u = v
52
were treated in
22.6.8. Finally, we give some diagrams which are due to B. CARL/B. MAUREY/
J. PUHL[ 11 : 'I 2
L+L
r 2
22.7.
Embedding Maps of Sobolev Spaces
22.7.1. Let l2 be a bounded open subset of the n-dimensional Euclidean space 9" having n sufficiently smooth boundary. For 2 2 0 we write 1 = 6 [A],where [A] is the greatest integer not exceeding I and 0 2 8 < 1. If 3. = 0, 1,2, ..., then the Sobolev space W#I) consists of all functions f defined
+
22. Operiltors in L,-Spaces
321
on Q such that the weak derivatives Df of order la1 2 I. exist and belong to L,(Q). I n the case where R 0, 1,2, it is assumed that, for all a wit,h1 . 1 2 [I.], we have
+
...
and
Then W#2) is called a 8obolev-8Zobodetzkij waee and a Holder space, respectively. Let us mention that W$2) becomes a Banach space with a suitable norm. The theory of these spaces is fully presented in [TRI]. 33.7.2. As a basic result we state Sobolev's
Theorem. Let A/n > l / u - l / v 2 0. Then W:(SZ) c L@). 22.7.3. For every operator ideal U the Sobolev limit order u,(a, II, v) is defined to be the infimum of all A > 0 such that the embedding map from Wt(i2) into LJQ) belongs to a. 22.7.4. Sobolev's limit order is closely related to the ordinary one. So, in order to obtain informations about embedding maps of function spaces it is enough to consider diagonal operators in sequence spaces. The non-trivial proof of the following L KONIU[l]. result is given by E T h e o r e m . Let
[a,A] be u quasi-normed operator ideal. l'hen
o,(%, u, v)/n = A(%,
26,
v)
+ l / u - I/@,
where 12 i s the dimension of Q.
22.7.5. Suppose that the bounded kernel K is infinitely differentiable on Q the eventual exception of the diagonal. Then, for certain u and v,
K , :f ( t ) -+
g(8) :=
SZ with
J IS - tl"(a-l)K(g,i?) f ( t ) dt
Q
with a > 0 defines a so-called weakly & t &ar
iittegrcrl operator from Lu(0)into
WQ). Special examples are the Riemann-Liou.viL?eoperators of fractional integrdion given bv
B, : f ( t ) + g ( s ) := -
J'
rb)0
where 0 5 s, t
(8 -
tp-1 f ( t ) at,
5 1.
22.7.6. The significance of Sobolev's limit order follows from t h e next T h e o r e m . Let U be an operatorideal. If a
> ua(%, u, v), then K , c 21(Lu(Q),L*(fl)). P
R e m a r k . The above condition is almost necessary and sufficient, since, conversely, a < u p , ~ ~ (u, U ,4 implies Ra 6? U(L,[O, 11, LJO, 11). 21 Pietsoh, bpcrat,or
332
Part 4. Basic Examples of Operator Ideals
22.8.
Notes
There is an extensive literature dealing with operators in L,-spaces. First of all we have to mention the pioneering papers of A. GROTHENDIECK [4] and J. LINDENSTRAUSS/A. PELCZYRSKI [l]. The coincidence of Hilbert-Schmidt operators and absolutely r-summing operators in [4] and A. PIETSCH [5]. Further results conHilbert spaces was proved by A. PILCZY~SKI cerning absolutely (T, p, q)-summing and ( T , p, 9)-nuclear operators in Hilbert spaces can be [2], G. BENXETT/V. GOOD MAN^. M. NEWMAN [l], B. CARL [I]. [2], [&I, found in G. EENNETT D. J. H. GARLIBG [3], Y. GORDON/D. J. LEWIS [3] and A. PIETSCH/H.TRIEBEL[l]. The main theorems about absolutely r-summing operators from L, into L, are taken from S. KWAPIE~~ [5], [6] and I?. SAPHAR [3]. See also G. EENHETT [l], B. CARL [2], [3], D. J. H. GARLING[5], A. PIETSCR [16], and A. E. TONG[l]. The definition of a limit order was given by A. PIETSCH [14]. We also refer to a famous paper of H. KONIQEl]. Recommendations for further reading:
[LIN], [LOT]. G. BAUMEACH/W. LINDE[l], G. BENNETT [4],B. CARLIB.MAUREY/J.PUHL[I], C. CLARK [I], J. A. COCJIRAN[I], Y. GORDON/D. R. LEWIS/J.R. RETHERFORD [l],[2], D. J. H. GARLIXG/ Y. GORDON[l], E. D. GLUSKIN [2], J. R. HOLUB[3], [5], J. S. HOWLAND 111, H. KONIG[2], [3], [4], J. L. KRWINE[2], S. K W A P I E[l], ~ [2], D. J. LEWIS121, J.LINDENSTRAU~~/H.P.RO~E THAL [I], 13. MAUREY[2], A. PEECZY~~SKI/W. SZLENR[l], G. PISIER [5], N. POPA[l], J. R. [ 2 ] , R. E. RIETZ[l], R. ROGGE [l], [2], H. P. ROSEXTHAL 141,[6], N. TOMCZAKRETHERFORD JAEQERMANX [l], A. E. TOXG[2], L. SCHWARTZ [l].
LUST Theory
23.
This cha.ptor is devoted to special kinds of absolutely summing, nuclear, and intsgral operators which play an important role in the structure theory of Banach spaces. The abbreviation LUST is derived from the phrase “local unconditional structure”.
23.1.
Absolutely z-Summing Operators
23.1.1. An operator X E B(E, F ) is called absolutely z-summing if these is a constant c 2 0 such that
for all finite families of elements zl, We put
...,xm E E and functionals bl, ...,bm E P’.
P,(S) := inf 6. The class of all absolutely t-sunzming operators is denoted by ?&.
23.1.2. The proof of the following result is straightforward. Theorem.
I?&,
P,] i s a normed operator ideal.
23.1.3. Analogously to 17.1.3 we have the Theorem. The normed operator ideal [PI, P,] is maximal.
23.1.4. Theorem. The normed operator ideal [?&,PI] is completely symmetnk. Proof. Obviously we have PP”’(fJ) = P,(X) for all elementary operators. Hence the assertion follows from the niaximality of and 8.7.12.
v,
23.1.5. The ideal of absolutely z-summing operators is rather small. Proposition. I f r 2 1, then [g,, P,] S [ ? P ( r , p , q ) , P ( r , p , q ) l * P r o o f . Let 8 E I;P,(E,I?). Since l/r‘
21 *
,
+ lip + l / q 2 1, we have
324
P u t 4. I3auir Examples of Operator Ideals
23.1.6. T h e o r e m . An operator S E B(E, F ) i s absolutely t-summing i f unCE mdy i f there exists a constant u 2 0 a d a probability ,u E W (UBp X UF,,) such that
I(%, a} (y", b)l
@'x, b)l 5 u
dp(u, y")
for all z C E and b E F'.
us. x UP..
Here we suppose that UBtand UFttare equipped with the weak E-topology curd 3"-toplogy, respectively. In th& case,
P,(S) = inf u. P r o o f . The sufficiency of the above conditioii is evident; cf. 17.3.2 and 17.4.2. Conversely, let S E ?&(E, P) and put u := P,(S). Take C(UETx Up*)' equipped with the weak C(UEpx UF*r)-topobgy. Then W(UE,x UFrr)is a compact convex subset. For any finite fanlilies of elements xl,. .., x,,, E E andfunctionals b,, . .., b,,, E F' the equation m
~ ( p:= ) 2 ~ ( S Xbi>l ,, - u 1
WE'
J
1(xi, a>iy", bi)l dp(u, y")
x Ute,
defines a continuous convex function @ on FV(Gfi,x UF.v).Choose ':y E UFtesuch that
a,
6 U,. and
If 6(uo,y,") denotes the Dirac measure at (h,y,"), then we have m Q(d(a0,
Y;))
X [I(Sxi, bi)i -
=
i(xi, uo) (YC,hi)\] 5 0 .
1
Since the collection 9 of all functions obtained in this way is concave, by E.4.2 there is po E W(UB.x UF-) such that @(yo)5 0 for all @ E 9 simultaneously. In particular, if @ is generated by the single fainilies (x)and ( b ) , it follows that
J
I(Sx. b)i -
UEfx
I(x, a>(Y", b)l dpo(a, 2/") 5 0 . t,
This completes the proof.
23.2.
a-Nuclear Operators
23.2.1. An operator S .
B(E,F ) is called u-sl;uclear if
N
S=
a i @ yi 1
with a,, %, ... E E' and y,, y2, in the operator norm.
...E F
such that the family
(ai @
yi) is summable
23. LUST Theory
325
We put W S ) := inf sup
K
I1
z.‘ I(z, u L >(yE,6)l: llzd 5 1, Wll 5 1
,
ahere the infimum is taken over all so-called o-nuclear representations. The class of all o-nuclear operators is denoted by 8,.
53.2.2. The proof of the following result is straightforward.
[a,,Nu]i s a normed gperator ideal. A sequence ( e , ) with e, + o is called a hyperorthogonal
Theorein.
23.2.3.
qpace U if its linear span is dense and if I(,\ 1L = 1 , 2 . ...
basis of the Banach
1 : l II: l
5 lrrlimplies 2’ Eiei 5
for
R e m a r k . Every hyperorthogonal basis is unconditional. Conversely, a Banach space with a n unconditional basis can be renormed such that the basis becomes hyperorthogonal; cf. [STS, pp. 558-5601.
23.2.4. The standard examples are given in the trivial P r o p o s i t i o n . The unit sequence basis of l p with 1 5 p
< co a i d
co i s hyper-
orthogonal. 23.2.5. T h e o r e m . An operator S E 2 ( E ,F ) is a-nuclear if and onhy i f there exids corn mutative diagrn m
.such that A E Q(E,U ) and Y ortliogonal basis.
Q( U , 3’). Here U i s s m e Banach space having a hyper-
Jn this mse, N U ( W
= inf
IIYII IIAll7
where the infimum i s taken over all possible factorizations. Proof. The sufficiency of the above condition is obvious, since 03
S = 2 A’fi @ Ye,, 1
where (ei) denotes the hyperorthogonal basis and coordinate functionals.
(Ii)is the sequence of corresponding
To check its necessity we take a o-nuclear representation m
S = 3 ai @ y i 1
326
Part 4. Basic Examplca of Operator Ideals
such that
Obscrve that 05
an := SUP
2,' I(%, ~ {nil
i( ~ ) ib)l: ,
l l ~ ld l 1, llall 5 1
tends to zero; cf. [PIE, p. 251. Using the method of 8.6.4 we can find that 1 2 el 2 ez 2 ... 2 0 and
+
05
zeb2 I(%,
U i ) (Yi,
(en)E co such
a)] I (1 4') WS) IIXI! Ilhll-
1
Let U be the collection of a11scalar sequences t = (ti)for which ( tigi*yi) is suniniable. Then U becomes a Banach space with the norm i'tl: := ~ v ~ ( t ~ g ; ~and y ~ ) the , uriit sequences form a hyperorthogonal basis. Define A E 2(E,U ) and Y E 2 ( U , F ) by m
A x := ((z, ai)) and Y(r7i):=
qiyi. 1
As usual we pnt p n ( E i , .-*, t n ,
tnti,
a * . )
:= ( E l ,
. . a ,
En,
0,..*).
Then
5 en+l lltll-
+
Hence 11-4 - PnAI:5 en+l(l E ) X&!3) ~ and /IT' - YE',,' 5 p3,Ll. 80 A E %(e, G) and Y E @ ( U ,F ) . In particular, we have //Ail 5 (1 E ) ? S,(S) and iiYl[ 5 1. Since S = Y A , the assertion is established.
+
23.2.6. As an immediate consequence of the preceding factorization propertoy we get tile
Theorein. T h e normed operutor ideal [?&, &] is ?ni?iitn.tl. 23.2.5. The next result is also obvious. Theorem. The normcd operdor ideal [Ru,Nu]is sytnmetric. R c m a r k . I t seems very likely that [nu, X u ] is L o t regular. 23.2.8. The ideal of a-nuclear opcrators is rather 1~1.,i 2 . TVe have the eyident i'roposition.
[R(r,p,q), N(,.,p.q)l S [a,, XI.
23. LUST Theory
327
23.2.9. For every operator S E g ( E , P)we put
[
?J:(X) := inf sup
c:
2 I(x, a$ (yj,a)!:
11
ilxli 5 I, llbIl 5 1 ,
;There the infimum is taken over all finite representations n
s = 2 a*0yi. 1
23.2.10. Proposition. rU: is a nmin on &. 23.2.11. Using the technique of 18.1.12 we can check the regularity of 1;.
Proposition. If S E g ( E , F ) , then N:(S) = X:(KpX). 23.2.12. Analogously to 18.1.14 we have the following estimates. L e m m a 1. L e t X
&(Eo,E ) and X E ?Jt,(E, P).Then
N : ( m 5 Nu(& IlXIl. Lc innin 2. Let S E %JE,F ) and B E &(F,Fo). TherL PI':(BS) 5 IIBII YAW ' 23.2.13. With the method of 18.4.5 we get the Thzorem. [%:, N,*]= [qZ, P,].
23.3.
o-Integral Operators
23.3.1. An operator
X E B(E, F ) is called o-integral if
it belongs to the nornied ideal
1%
I,] := [%", 8,1mas. 23.3.2. As a counterpart of the preceding definition from 8.7.15 we get the Theorem. [%, Nu]= [3,,Iulmin. 23.3.3. The following result is analogous to 19.2.5 and 19.3.6. Lemma. L e t V be an order cornpZete Bunach lattice. I f X E $(E, P), then pli:(S) = IiSll. Proof. First we check the assertion for those operators admitting a finite representation n
So = 2 aoi @ h4 1
such that hi A hi = o if i
+ j . Let 2 c E and g E
I.&, aoi) (hi, g) = I(%,
aoi>(7~,,
g)l
and
Then n
zI(%, 1
n
Uoi)
(hi, g)l = 2 ' &(XL.,not>(hi, 1
s>
V'. Choose A1,
l&i
=
1.
...,1, E X
with
338
Part 4. Basic Examples of Operetor Ideals
Now let S E g ( E , F ) be arbitrary. Take some finite representation in
S =
ak Q f k 1
rn
2' IIaki; 5 1. Siniulating approximation by simple functions, given E > 0, we 1 n can find f o k = tikhisuch that Ilfk - fokll 5 8 and hi A hi = o if i + j ; [LAC, p- 'i]
with
i=l
and [SAF,, p. 1371. P u t nr
80
:= 3 a k
8f o k .
1
Then m
n
So =
G~@ hi with
aoi :=
i=l
&a&.
k=l
Hence m
11s
-
5 N:(S - 80)5
1
ll%ll llfk - f0dl
We now get
N!(S) 5 q ( S 0 ) 4-N:(s - So) 5
llfloll
+
&
E.
I IlSll
+ 2.7.
This proves that N:(S) 5 /lSil. The converse inequality is evident.
23.3.4. Theorem. An operator S E B(E, F ) i s o-integral if and only if there exists a commutatiee diagram
su,ch that A E e ( E , V ) and Y E e ( V ,F'). Here V i s some (order complete) Banach lnttice.
In this ease,
Ids) = inf llyil i1-41, where the infimum .is take78 over all pmsible faetorizations. Proof. Obviously every Banach space with a hyperorthogonal basis is a Banach lattice in its natural ordering. Now it follows by ultraproduct techniques that KFS factors t*hrougha Banach lattice 7, if X E S,(E, F ) . Since KFS = YA implies KFS = (Kp)' Y"h',A, we may replace V by its order complete bidual V". Conversely, let us suppose that KFS = YA with A E 2(E,V ) and Y E B(V, P') such that the Banach lattive 7' is order complete. Let X E g(Eo,E ) and E $(p,PO). By 23.3.3 we have x!(BSx) = N:(pKFSX) I IIB"Yi1 * ( A X ) 5 IPII IIYII IIAIl IIXll. Hence S E 3,(E,
F)and I,(S) 5 IIYII 114 .11.
23. LUST Theory
329
23.3.6. A4nalogo~isly to 19.1.9 we have the
Lemma 1. Let X E g(Eo,E ) and S E 3 J E , P).Tlien
N:(SX) 5 I,(S) I]X]l. Proof. Given E > 0, we choose a factorization hrFS= Y A such that A E B(E, V ) , Y E jJ( T’, F”),and I/ YIl JIAll5 (1 + E ) 1,(8), where P is an order complete Bttnach lattice. Then 23.2.11 and 23.3.3 yield
s:(Sx; = Y:(K*,Sx)5 ilKFl-:!3 ! ( A X ) 5 IIYII IIAXIJI (1
+
E)
I@) IlXll.
This proves the assertion. For completeness we also formulate the dual estimate.
Lemma 2. Let S E 3 , ( E , F )and B E g ( F , Po).Then
qWS) s IlBll U W 23.3.6. Finally, we mention an iiiiportsnt formula which follows from 33.2.13. Theorem. [b:, P:]
= [3,, I,] and
[a,’,1 3 = [?&, P,].
23.3.5. -4 Banach space is said to have LUST, that means local mconclitional strucP.GORDOWand D. R. LEWIS), if it beloiigs to the space ideal
ture (in the sence of
I, := Space (3,). Remark. By 23.2.3 and 23.2.5 every Banach space with an unconditional basis possesses local unconditional structure. Moreover, we know from 23.3.4 that the sarne property even holds for all Banach lattices.
23.4.
Multiplication Theorems
23.4.1. Theorem.
[PI,P,] o [So,I,] s [B1,L,].
Proof. First suppose that T E g(E, F) and S E !@,(P, G ) . Given
E
> 0, choose a
23. LUST Theory
23.5.
331
Notes
The reader is warned that our defiaition of Banach spaces with local unconditional structure is a slight modification of t.hat given by E. DUBINSKY/~. PELCZYASKI/H. P. ROSENTHAL [i]. Nost results of this chapter can be implicitely found in P.GORDOK/D. R. LEWIS[l], [2], [3]. Recommendations for further reading:
[SEM,, exp. 24, 251
T. FIQIEL/W.B. JOHNSON/L. TZAFRIRI [l], T. FIGIEL/G.PISIER [i], W. B. JOHNSON [3], W. 3.JOHNSOX/L. TZAFRIRI [l], D. R. LEWIS[3], G. PISIER [6], J. R. RETHERFORD [2]. See also t,he bibliographical notes of Chapter 28.
24.
Decomposing Operators
The famous Dunford-Pcttis-Phillips theorem states that every weakly compact operator X from Ll(Q,p) into any Banach space E can be represented in the foriii Xf =
J x(0)f ( w ) 440)
for all
f E L,(Q,y )'
0
where x is a p-measurable E-valued function defined on the probability space (Q, p). A Banach space E is said to have the Radon-Nikodym property if this result remains true for all operators. I n this chapter we deal with thc ideal 1 ' ) of so-called Radon-Nikodym operators is that of decomposing and give its main properties. The dual ideal 0 = '1)'"' operators. We also introduce the ideal Q, of p-decomposing operators with 0 < p 5 oc. It turns out that Q, = p y and 8, = Q y for 1 < p < 03. Using the concepts described above we can improve some multiplication theorems for absolutely summing and integral operators. In particular, Grothendieck's formula ?lBo 3 = 92 is established.
24.1.
Measurable Vector Functions
24.1.1. In what follows ( Q , y ) always denotes any probability space. By an Ecalued function x we mean a map from SZ into an arbitrary Banach space E. 24.1.2. An E-valued function x is said t o be p-sinaple if n
Xih,(Cu),
x(w) = I
...,xn E E and ...,Qn of 9.
where q, subsets Q1,
h,, ...,h, are characteristic functions of y-measurable
24.1.3. An E-valued function 2 is called y-measurable if there exists a sequence of p-simplo E-vtllud functiocs a, such that x(to) = l k xn(o)ahnost everywhere. n
24.1.4. Let us mention the trivial Lemma. For every p-measurable E-valued function y-measurable, rts well.
d:
the scalar funclion
I@(.)\\i s
24.1.5. Now an important characterization of p-measurable vector functions is stated. For proofs mo refer t o [DIU, p. 421, [DUN, p. 1491, and [IOX, p. 721. Theorem. An E-valued function x i s ,u-nreasurable i f and only if the f o l l o w i q ccnditions are satisfied: iL (1)There exists a separable subspace A4 such that x(w) E iW almost everywhere. (2) All scalar functions (x(.), u) with a E E' are p-measurable.
336
Part 5. Applications
41.1.6. Lemma. Let x be cc fi-measurable E-valued function. Suppose that for all a E UE, we have I(z(o),a ) / p almost everywhere. Then 112(0~5 )!1p wlinost everywhere.
Proof. First we observe that alniost all values of x belong to some separablc subspace M of E. Let ( x n ) be a countable dense subset of M . Furthermore, pick a, 5 UE#with (xn,ajL)= Ilz,J/. Then = 1,2,
llzll = sup (i(x, a,)! :
...)
for all x E X.
Finally, we can find a p-null set A' such that I(z(o), a,)l
2 p for all w E Q \ A'
and
12
= 1, 2,
...
This yields Ilz(m)ll (= e almost everywhere. 2 1.1.7. I n what follows we do not distinguish between p-measurable E-valued functions which coincide alniost everywhere. The collection of these equivalence classes is denoted by L,(E, Q, ,u). Let
Ilzllo:= inf ( p )= 0: p(w E 12: Ilx(w)ll > e) 5 e j . Using standard techniques we get the T h e o r e m . Lo(E,0, p ) is a complete metric Zineur r p c c with the P-norm l~.llo.
21.1.8. If z is any p-simple E-valued function of the form 7k
a(o)= 2 x i k i ( o ) , 1
the11
1z ( w ) dp(c0) 2 Zip(Qi) :=
?I
1
is well-defined: cf. 24.1.2.
24.1.9. An E-valued function x is called p-integrable if there exists a sequence of p-simple E-valued functions x, such that lim IIz - zn/io = 0 and II
hm.nn
j
//Z,,~(O>)
n
-
x,(o)ll d p ( o ) = 0 .
Then the Bochner integral is well-defined by
J z ( w ) d,u(w) := lim J x,,(to) d,u(to). n
'I
D
94.1.10. By [DIU, p. 451 we have the Lemma. An E-valued functim t is p-inteyrcrble if md only if it is p-tueu8urable and dP(0) < 60-
J Il=(c.)l 12
24. Decomposing Operators
24.2.
337
Radon-Nikodym Operators
24.2.1. An operator X E f?(L,(Q,p), E) is called r@ht decomposable if there exists a ,u-measurable E-valued function x such that
X f = J x(w)f ( w )++J)
for all f E L,(Q, p ) .
R
Then x is said to be the kernel of X . 24.2.2. The following lemma can be derived froin 24.1.6.
Lemma. The kernel x of every right decomposable operator X E 2(Ll(12,p), E ) i s unique almost everywhere. Horeover,
llXIl
= ess-sup ( l ~ x ( w ) w ~ lE: Q ) .
24.2.3. We next state the famous Dunford-Pettis theorem a proof of which can be found in [BOU,, chap. VI, p. 461, [DIE, p. 2251, and [DUN, p. 5031. Theorem. Let E be a B a M space such that E' is separable. Then every operator X E s(L,(Q,p ) , E') is right decomposable. 36.2.4. We are now able to check the deep Theorem. Let E be a reflexive Banach p e . Then every operator 3 E f?(Ll(Q,p), E') is right decomposable.
Proof. We know from 3.2.9 that L,(Q, p ) has the Dunford-Pettis property. So every operator X E f$(L,(O, p ) , E ) is completely continuous. Observe that the embedding map I from L , ( Q , p ) into L,(Q,p) is weakly compact. Thus, by 3.1.3, the product X I must be compact. Consequently 1M := lM(X1)is a separable subspace of E . Since L,(Q, p) is dense in L,(Q, p), we also have M ( X ) = M . So X acts from Ll(Q, p ) into M = MI', and we can apply 24.2.3.
24.2.5. As an immediate consequence of 2.4.3 and of the preceding result we get the Dunford-Pettis-Phillips Theorem. Every operator X E !Xl(L,(Q,p), E ) is right decomposable. Remark. A direct proof is given in [ION, p. 891. See also [DIU, p. 751.
f-1.2.6. Let S 2 ( E , F ) . Then S is called a Radon-Nikodynz operntor if S X is right decomposable whenever X E B(L,(Q,p), E ) . The class of all Radon-Nikodym operators is denoted by g. 24.2.1. Theorem. 9 .is a closed operator d e a l .
Proof. It can be easily seen that m
of operators S, E g ( E , P)with are kernels yn such that
s , X f = J y,(w) n
22 Pietsch. Operator
9 is an operator ideal. We now take a sequence
llSnli< OQ. Given X E 2(L1(D,p), E),then there
1
* f(w) dp(w)
for all
f E L,W, P I .
338
Part 5. Applications
By 24.2.2 we have I]yn(w)]] 5 l]SnXilalmost everywhere. Since ca
m
2' IlgAw)!I 5 Z IIfL!' !!XiI, 1 1 by setting Qi,
Y(0) :=
z
Yn(c.)
1
we get a p-measurable P-valued function which is defined almost everywhere. So we m
have found a kernel y of the operator SX, where S:=ZS,,. 1 S E g ( E , F).Hence the operator ideal 9 is closed.
This proves that
22.2.8. Theorem. The operator ideal 'I) is injective. Proof. Let S E %(E,P).Then, given X E B(Ll(s2,p), E ) , there exists a kernel y such that
S X = ~ J g ( w ) /(oj) d,u(w) for all f E L,(SZ,p ) . R
-
Put B := M ( S X ) . Applying 24.1.6 to the FIN-valued function Q;y y(w) E d l ( 8 X ) almost everywhere. This proves the injectivity of 9.
we see that
24.2.9. A Banach space belonging to Y := Space ('I)) is said to have the RadoaNikodym property. Remark. The thcory of these Banach spaces is presented in [DIE] and [DIU].
242.10. Proposit ion. The B a m c h space 1, possesses the Radon-Nikodym property. Proof. The assertion follows from 24.2.3, since 1, is a separable dual Banach apace.
24.2.11. Proposition. The Banuch spaces Ll[O, 11 and co fail to have the R&ANikodym property. IZ
Proof. Let I be the countable set of all indices (k,n ) with k Porin the intervals
= 1, 2,
...
= 1,
...,2" and
and denote the corresponding characteristic fiinctions by hkn. Then the equation Xf:= ( f , hkn)defines an operator S E B(L,[O, 11, co(I)).Let us assume that L,[O,11 E Y or co(I)E Y. Then X is right decomposable, and we can find a co(l)-valuedliernci X. Obviously z(t)= (hkR(t))almost everywhere. On the other hand, we always have (hkn(t)) 6 since card ((k,n):hkn(t)= 1)= No for all t E (0, 11. This contradicticjn implies L,[O, 11 ( Y and c o ( l ) Y. Clearly, %(I) and co are isoiuorphic.
24.2.12. Proposition. !ll3 c 'I). Proof. The inclusion follows from 24.2.5. Moreover, we have lI E Y \ W.
24. Decomposing Operators
339
24.2.13. Proposition. QJ c 8. Proof. Suppose that some S E QJ(E, F ) is not unconditionally summing. Then by 1.7.3 there exists X E B(co, E) for which SX is an injection. Now the injectivity of QJ implies that the identity map of c,, belongs to B. This is a contradiction.
24.2.14. Proposition. The operator deals ?Band QJ are incomparable. Proof. Let X be t,he operator constructed in 24.2.11. Then X E B \ QJ.On the for ll all illfinite dimensional reflexive Banach spaces E . other hand, we have Ix E B \ ?
24.3.
Vector Measures
be any a-algebra given on a set 9. A map m from $' 3 into a Banach 24.3.1. Let space E is called a vector measure if
for every sequence of disjoint subsets B,, B,,
..- E $' 3.
34.3.2. The variation po of a vector measure m is defined by
The supremum is taken over all finite families of disjoint subsets B,, belonging to %.
...,B, of B
24.3.3. The proof of the following lemma can be found in [DIN, pp. 32-35]. Lemma. The variation of every vector meamre is a smlar measure which is, however, 7zeces8arilyfinite.
not
24.3.4. We now consider a vector measure m and a probability p, both defined on the same a-algebra 8.Then wz is called p-continuous if p ( B ) = 0 implies m ( B ) = o for B E 8.The E-valued measure m is said to be p-differentiable if there exists a p-integrP+bleE-valued function x such that
m ( ~=)J z(w)d,u(o) for all B
c B.
B
Then x is called the Radon-Nikodym dericative of m. 24.3.5. Let m he any E-valued measure. Then the P-valued ineasurc Stti, where 8 E B(E,F ) , is defined by ( S m ) (B):= X[rn(B)]for B 8.
21.3.6. Every p-differentiable E-valued measure m is p-continuous and has finite variation. The fact that the converse stakment is false gwve rise to define RadonNikodyin operators; cf. [DIU, p.501. Theorem. An operator X E 2 ( E , F ) is a a d o n - N i k o d y m operator i f and only i f it ?naps every p-continuous E-valued memure m with finite variation h t o a p-differcntiable F-valued measure Xm.
.
22*
340
Part 5. Applications
Proof. Given X E B(Ll(Q,p), E ) , then m ( B ):= X(hB) for B E % defines a y-continuous E-valued measure m with finite variation. Let us now assume that S possesses the property mentioned above. Then there exists a derivative y of am. This means that = Jy(w)
SX(ILB)
~ B ( w dy(w) )
for all B E 8.
0
Consequently we get
SXf =
J Y(0)f ( w ) 4 4 w )
D
for all p-simple functions f. By continuity the same formula holds for all f E L,(Q, p). This proves that 8 E g ( E , F ) . To check the converse implication we consider a p-continuous E-valued measure m with finite variation po. Then there exists an operator X E B(Ll(12,h),E ) uniquely defined by the condition X(hg) = m ( B ) for all B E %. If 8 E g ( E , F ) , then we can find a p-measurable F-valued kernel yosuch t,hat
8x1= J ? l o b ) fb)dpo(w)
for all f E LdQ, Po)
-
Q
Since the p-continuity is transmitted from m to po, according to the classical RadonNikodym theorem, there is a y-integrable scalar function g 2 0 such that ,tAo(~)=
J g(w) d p ( w )
for all B E B .
I3
Consequently, by putting y := gy,, we get Xm(B) = S X ( ~ B ) = J yo(@) +,(to) B
=
1y(o) dp(cr>).
B
s o y is the derivative of Sm. This proves that every Radon-Nikodym operator has the required property.
24.4.
Decomposing Operators
24.4.1. An operator A 6 B(E, L,(S, p)) is called left decomposable if there exists a y-measurable E-valued function a such that
Ax
= (2,
a(.)) for all x E E .
Then a is said to be the kernel of A . 24.4.2. An operator S 6 B(E, F ) is called decomposing if BS is left decomposable whenever B E f?(F,L&2, p)). The class of all decomposing operators is denoted by a.
24. Decomposing Operators
341
TZ = 9"'. P r o o f . Let S E Q(E,P).If Y E B(L,(sZ, p ) , F'), then Y'K,S is left decomposable.
24.4.3. Theorem.
For x E E and f E &(Q, p ) we have (2,S'Yf) = !Y'Kp!h, f
J (x, ~ ( w )f ) (
)=
~d )p ( ~ ) ,
Q
where a is the corrcsponding kernel. The above equation yields
S ' Y ~= J a ( o ) f ( o ) d,p(w) for a11 f E L,(Q. p). Q
This proves that x't'
g(F',El). Hence D
gaual.
Conversely, let X E (I)*""(E,F ) . If B E g ( P ,L,(O, p)),then S'B'KL1is left decomposable. For x E E and f f L,(D, p ) we have
I(%,N o ) )
{B~X f ) ,= (x,S'B'K,,f) =
f(0))
~Aw),
s1
where a is the corresponding kernel. The above equation yields BSx = (2,a(.))for all x E E. This proves t8hatS E Q(E, P).Hence gd*' Q. 24.4.4. As a counterpart of the preceding result we formulate the
P r o p o s i t i o n . Qdual c 9. ( ~ ) d u a l ) d u n5 l Oreg = g. In order to = (Ddual implies Qdual Proof. Clearly show that Qdw1 + 9 we consider the related space ideals. By 24.2.10 we have I , E Y. On the other hand, the assumption I , E ad"'implies 1'; E Y. Since 1'; and LJO, 11" are isomorphic [SED, p. 4801, the injectivity of Y yields Ll[O, 11 E Y, which is a contradiction by 24.2.11.
24.5.
p-Decomposing Operators
243.1. Let' 0 < p m. An operator A f B(E,L J 9 , p ) ) is called left decomposable if there exists a p-measurable El-valued function a such that
A x = (2,a(.)) for all x E E . Then a is said to be the kernel of A . 24.5.2. Lemma. The kernel i.s unique almost eoeryuhere.
of
every left decomposable operutor A E B(E;L,(Q, p ) )
Proof. Obviously it is enough to show that. m y kernel a of the zero operator vanishes almost everywhere. First we observe that almost all values of a belong to some separable subspace M of E'. Let (a,) be a countable dense subset of M and choose x, E U, such that 2 i(xn, a,)[ 2 Ilu,ll. Then llallO
:= sup (I(x,, .}!:
n = 1, 2,
...I
342
Part 5. Applications
defines an equivalent norm on M . More precisely, we have 2 lla/lo2 /lull 2 jjaljOfor all a E M . Take a p-null set N such that (zn,~ ( w ) = ) 0 for all
OJ
E s;! \ N
and n
=
1,2, ...
Hence l l ~ ( a ) )= //~ 0 almost everywhere. This proves the assertion. 24.5.3. Let 0 < p 5 00. An operator S E B(E,F ) i3 called p-decomposing if BS is left. decomposable whenever B f E(P,Lp(s;!, p)).
The class of all p-deconiposing operators is denoted by
Qp.
Remark. Obviously, we have Q = Q,. 24.5.4. We formulate without proof the trivial
Theorem. Q p is an operator ideal. 24.5.5. Using the method of 24.2.8 we get the
Theorem. The operator ideal Q p is surjective. 24.5.6. Theorem. If 1
p 5 00, then the operutor ideal Q, i s regular.
Proof. The regularity of 0, follows from 4.5.6 and 24.4.3. Let S E Q'dig(E,P)and 1 < p < 00. Then, given B c B(F, Lp(Q,p)), by 1.5.5 therr exists BzE B(F", L,(D, p ) ) such that B = B7KF.Consequently BS = BT(KFS) is left decomposable and therefore S E Qp(E,F ) . This proves that Q, = QFg. To treat the case p = 1 we need the fact that there is a canonical surjection Q from L,(D, p)" onto L,(B, p ) which can be defined by the help of Lebesgue's decomposition; cf. [BOU,, chap. V, p. 611. Then the left decomposibility of BS folloas from BS = (QB")(K,S). Remark. It seeins to be unknowii whether or not the operator ideal 0,is also regular for 0 < p < 1. 24.5.7. Theorem.
Qp
s FpdpU8l.
Proof. Let S E Qp(E,F),Then for every 6 E E(F, Lp[O,11) there exists a unique Lebesgue-measurable E'-valued kernel a such that BSrr = (x,a(.))for all x E E. Clearly B +u defines a linear map K froiii B(F, Lp[O,11) into Lo(E',[0, 11). We noTs consider a sequence of operators B,, B,, E E(F, L,[O, 11) converging to some operator B f E(F, Lp[O,11). Furthermore, let us assume that the sequence of corresponding kernels a,, C I ~ ., .. tends t o some vector function a E Lo@',[0, 13). Then
...
= Lp-lim(2,un(.)) = Lp-limBnSx = BSx. Lo-lim (r,an(.)) n
(I
11
On the other hand, by hypothesis we have Lo-lini(2,an(.))= (x,a ( . ) ) . n
Hence u is the kernel of BS. This proves that the map K is closed arid therefore continuous. Choose 6 > 0 such that \\I311 5 6 iniplies I/allo5 1/2. We claim that
24. Decomposing Operators
343
for all finite families of functionals b,, ...,b, E F‘. In order to verify (*) we inay assume that wp(bi)= S and I[S‘bi[l> 0. Put l i
II~‘bilP
+ + IIfJ’bmll”
:=
IlS’blllp and form the intervals
..*
If h,, ...,h, are the corresponding characteristic functions, t,hen bi @ hi
B := defines an operator B
B(P,L,[0, 11) with ijB[j= 6. Since
m
1
is the kernel of BS, it follows that /\ullo5 lj2. On the other hand, we have that i:n(t)ll= lp(8’bi)for all t E (0,11. So lp(8’bi)> 1/3 would iinply I!alio> 1/2, which is a contradiction. Therefore (*) holds, and t,his means that S’ E !J&,(F’,El). Consequently
a, 5 qy.
31.5.8. To prove the main result of this section we need thc following Lemma. Let 1 p < 00 and A E p y ( E , L,(Q, ,LA)).Then there emkts f i L,(Q, p ) such that \Ax1 5 f
for all
2
E
U,.
I n other terms, A maps the closed unit ball U , into an order bounded subwt of Lp(!2)p). Proof. Take any finite p-measurable decomposition (Ql,
..., Q m ) of B such that
p(Qi) > 0, and denote the family of corresponding characteristic functions by (hl, .,hm).Defineu,, u, E Lp(Q,p) and cl, v,,, E L,.(Q, y ) by ui := ,u(Qi)-’%i and vi := , U ( B ~ ) - ~ ’Then %~.
...,
..
...,
m
L := 1z’j 0ui 1
is an operator in LJQ,,u);of. 19.3.5. It follows from wP(vi) 5 1 that lp(A’vi)5 Ppd‘(A). SO w
fL
:=
2 [IA’viIiui 1
fulfils llfLllp 5 P;mi(A).-Since
344
Part 5. Applications
..
Consequently max (JLAzl:,I, ., ILAs,l) 5 fJ for all finite families of elements x,, ..., x,,E U,. This yields llniax (\LAX,!, ..., ~LAx,l)ll,2 P y ( A ) . Observe that, LAz tends to Ax, if (9,; ...,9,) ranges over all finite p-measurable decompositions of 9 ;cf. 19.3.5. So, passing to the limit, we have
..
Since the family of all functions max (IAxll, ., IAx,]) is directed upwards, according to [BOU,, chap. IV, p. 1371, there exists a supreinum f E Lp(9,p). Clearly f has the required property.
We are now able to establish the fundamental Theorem. If 1 < p
< M, tkcn. Q p
=
,y
and
Qiml= Qp.
Proof. Let S E Q,(E, 3') and X E O(E', Lp(9,p)). According to 17.3.11 without loss of generality we may suppose that B' is reflexive. By the preceding lemma there exists a positive function f E L p ( 9 ,p ) such that JXS'b/5 f for all zi E UFt.Hence Yob := XS'b/f defines an operator Y o B(F', L,(O, p)). It follows froin IF E 9 that IF, E Q. Thus we can find a p-measurable F-valued kernel yo such that P,b = (yo(,),b) for all b 6 li". Hence XS'b = (g(.),b) for all b E F', where y := /yo. Therefore S' is p-decomposing. This proves that Q, & Qy. As proved in 24.5.7 we also have Q p & So we get
vy.
!ppE Q y l & ( v y ) d u a l s vp and dual Q,dual c =(Qp
dual
c
=DpSPyl.
24.5.9. As a counterpart to the preceding theorem we have the Proposition. Q,
+ Py'.
Proof. The canonical map J , from C[O, 13 into L,[O, 11 is 1-integral. Hence J , E 3p1 5 gp$".We now assume that J1E Q1. Then J1has a Lebesgue-measurable C[O, 11'-valued kernel j,. Form the polynomials pn(t):= tn for n = 0, 1, ..., and choose a Lebesguc-null set N such Ohat
pt2(t) = (p?87j1(t)) for au. E
Lo, 1'
\N*
Sccording to Weierstrass's approximation theorem the above equat.ion holds for all functions f E C[O, 13. This means that j,(t) = 8(t) for all t E [0, 13 \ N .
However, there does not, exist any separable subspace M of C[O, 11' contailling almost all Dirac measures 8(t). This contradiction proves that J , 6 Q,. 24.5.10. Finally, we state the trivial Proposition. I/ p1 5 p2,then Qpl E Q,,. Remark. Using a famous theorem of E. M. NIKISHIN [i] it can be shown that the deal Q, with 0 < p < 1 does not depend on the parameter p .
24. Decomposing Operators
24.6.
345
Multiplication Theorems
24.6.1. I n order t o prove the basic result of this section we need a preliminary Lemma. Let (0, p ) be any probability space and 1 5 r < 00. Then for every left decomposable operator iZ E f?(E,L,(.Q, p ) ) the product 1,A is r-nuclear such that S,(I,A) 5 114 .11. Here I , denotes the embedding map from L&2, p ) into &(0, ,u). Proof. First we treat the special case in which the kernel a of A has a countable a(o)= aj}.Without loss of generality we may assume image (ai).P u t Qi := ( w E 0: that p(Sj)> 0. Let f i := ,u(Oi)-l/%i,where hi denotes the charact,eristio function of R,. Ther! 'x1
1,A
=
2,' ,u(R~)~!' C C ~S J fi. 1
It follows from
lr(p(Ot)l'r)= 1 , wm(ai)=r llAll, and wrt(fi)= 1 , that I,A is an r-nuclear operator with N,(I,A) 5 1 1 8 1 1 . We now come t o the general case. Since almost all values of the kernel a belong to some separable subspace H of E', we can find B sequence of p-measurable Elvalued functions a, possessing a countable image such that ess-sup (Ila(co) - an(co)i/: w 0) l/n. P u t A,z := (x,an(.))for all
5
E E. Then we have A = /l.ll-lim A,. Moreover, I
N,(I,A, - IrAn)5 llAm- A,li implies that (I,A,) is a n X,-Cauchy sequence. Since I,A is the only possible limit, we get I,A E a t , ( E ,L,(Q, p)) and
N,(I,A) = lim Nt,(I,A,) 5 lim liAnlI = IIAl]. n
n
We are now able to establish the fundamental Theorem. Let 1 2 r P r ,
< 00. Then
I,] 0 [a.Il-lil S
[a,,K P g .
P r o o f . Let T E Q(E,F ) and S E &(F, G ) . Given E > 0, by 19.2.6 we can find a factorization K,S = ZI,B such that IiZIl /IB// (1 c) Ir(8). Since BT is left decomposable, the preceding lemma yields KGST = Z ( I , B T ) E %,(E, G") and
+
Nr(KGfiF) 5 11211 Br(IrBT) 5 llZllllsrll 2 (1
+ &)
lITll.
This proves that S1' E ?J2Fg(E,(2) and lKf""(ST)5 I,(&')l!2'l.
24.6.2. As a consequence we get the famous Grothendieck Theorem.
[m, 11.111 o [Z, I] = [n,N].
Proof. By 2.4.3 every S E m ( F , G ) factors through some reflexive Banacli , and 2 E %(Go,G ) space Go. More precisely, given E > 0, we can find SoE f ? ( P Go) such that S = 28, and liZI[llSoll5 (1 E ) IISII. Now 24.2.12 yields Sh E gad((&, P'). So, by 24.4.3, we hare 8; E Q((2&3"). Let T E 3 ( E , F). Then T' E Z(F', E') and
+
:%46
Part 5. Applications
therefore T‘S;
%(GA, E’) as well as N(1”SA) 5 I(T’) IiSblI. Csing the reflexivity
of Go we get ST = Z(S,,T) E R(E, G) and
N(ST) 5
llzll N(SoT) i IIZlI IISoIl I(T) 2 (1
This proves that [!& 8.6.4.
11.11]
o [3,I]
[a,N].
+
E)
IiSII I(T).
The converse inclusion is evident by
R e m a r k . Let us niention that a n operator S E B(E,F ) belongs to 9 if and only if S-Y is nuclear for all strongly integral operators X E B(Eo,E ) ; cf. [DIU, p.1751.
24.6.3. We next state soine k i d of Grothendieck’s formula for r-integral operators. P r o p o s i t i o n . Let 1 4
T
< m. Then
[R,11.111 0 P r , 1 7 1 = [%,> 5 7 1 . Proof. Suppose that T E 3,(E, F ) and S E R(F, G). According t o 19.2.6 there exists a factorization KFT : YI,A such that !lYll llAij 5 (1 t F ) I,(!Z’). We have S7 %(F”, G). Since L,(Q,,u)has the approximation property, the operator S7Y is e v m approxiinable. So 19.1.10 yields ST = (S”Y) (IJ) E %,(E, G) a n d
S,(LSY)5
llsq ilAlj 5 (1 + 8) I’SIj I,(T).
r ,
I his proves that
[R,ll.lll 0 P,>I,] E [anKI. The converse inclnsion follows from 19.2.2. R e m a r k . Clearly
[m,ii.ii]
o [3,,I,]
4[a,,S,] for t
< r < 00.
24.6.4. We are now ready to improve 20.2.4. T h e o r e m . Let l / r
+ l/s = 1,‘p5 1 cxnd 1
: rafltxive Banach space. If X :3,(B’, G), then 24.6.1 iinplies K,SY E %,(Po,G“) a d X,(K,SY) 5 I,(S) IIY~~. Thrrefoie KGST == (K,SY) To E ?Rp(E,G”) and
Np(K,ST) 5 S,(K,SY) P,(?‘o) 5 (1
-c1
p:ores
i’z-:tig
+ F) I,(#) Ps(7’).
p).
to t!ie iiijective Iinlls froin (3) we get (1).
Let 7’ S s ( E , P ) and S I3 Y 3 ( F ,G). Then XT E 3FJ(F”, G) and 1;’ (&IT) = VJ(S). By 13.2.6 there is a factorization K,T = Y1,A with IlYll IlAll 5 (1 F ) Is(T). Since L6(l2,p ) is reflexivc, the operator Y is decomposing. So the injective version
+
24. Decomposing Operators
347
of 24.6.1 (theorem), namely
[Sr,Irlinj 0 [Q, 11.111 G [!It,. hTrlin', yields STY %F(Ls(s2,p), G ) and S:"j(S7Y)5 p ( S )llYIl. By 19.2.16 we can find a factorization S"Y = ZSo, where doE 3y(Ls(s2, p), Go),Z E A(Go,G),and
llZll I:"j(S0)5 (1 + e ) I:"j(S)IjY11. Altogether we get the diagrmi :
~,(-Q,
r ) T & ( Q ,r ) T . a o
Then S0IJ E S J E , Go) and Ip(#,,18A)5 I:"J(SO) llA/l. Finally, it follows from the preceding proposit'ion that ST = Z(SoIsA)E ?JIp(E,G) and
N,(ST) 5 IlZll &(SOLA) 5 (1
5 (1
+
C)Z
+ ).
1 3 4 IlYll
ll4
I:"?(#) 18(T).
This proves (2).
24.6.6. As a special case of the prtwding result we formulate the Theorem. [?&,PJ2 E [!It,N].
24.7.
Notes
Therc is an extensive list of papers dealing with representations of operators in function J. PETTIS[l], I. AT. GELFAXD spaces. We only mention the classical a o r k of N. DUNFORD/B. 111, and R. S. PHILLPS [i]. A full presentation and further references may be found in [DUN, pp. 489-5111. The theory of Radon-Nikodym operators has been developed by W.TJNDE [3] and 0. J. REINOV[i]. For further informattions the reader is referred t o i~ survoy paper of J. DIEsTa/J. J. Urn, [l] and the monographs [DIE] and [DIU]. Tii? coricept of a p-decompocf. [BAD, exp. I?$ Pitrther contributions sable operator was introduced by A. BAUBIKIAX; d r C due to S. KWAPIER [6], P. SAPHAR133, and L. SCHWAETZ [2], [3]. Th- most striking multiplication theorem of this chapter first appeared in [GRO. chap. 3. p. 1321. Other important results are taken f r o i A. PERSSOX 121. Becoxurnendations for furtlirr reading:
[DIN], [ION], [SEn/r,, annex I], [SElil,. csp. Oj. [SEA&, exp. -1-6.51, [SEi\I,. esp. 11, [swal,
[TAR]. S. BOCHNER [l], J. DIESTEL[i], V. K. K O ~ T K O[if, V [2], W. LINDE[B], 131, J. VON KEU[I], A. PIETSCH [4], [9], J. W. RICE[l], R. ROGGE [ I ] , C. SWARTZ [Z],G. I. TABGONSKI [I], E.THOMAS [I], [2], [3], J . WEIDNANN[l], J. WLOKA[l], [a], T. K. TVom 111, [2], [3].
26.
Radoriifying Operators
‘The concept of a cylindrical probability on special linear topological Hausdorff in 1933. Further progress is due to I. ill. spaces was created by A. N. KOLMOGOROV GELFANDand his fanioiis school. However, the final version of this important theory was given by L. SCHWARTZ. In general, a cylindrical probability defined on a Banach space may fail to be o-additive. So the question arises “What operators transform certain cylindrical probabilities in such a way that their images have better properties?” These considerations lead to the concept of a radonifying operator.
In the following we give a sketch of Schwarts’s theory which is presented in his “SBrnin&ir~s’~ from 1969 on. As a climax we discuss the Wiener probability appearing in the theory of Brownian motion. This probability is defined to be the image of some normalized cylindrical Gauss probability with respect to the integration operator. 26.1.
Probabilities on Topological Hausdorff Spaces
26.1.1. Let T be it topological Hausdorff space. Then @(T) denotes the Borel a-algebra genemted by the open subsets. Moreover,R(T)is thc collection of all compa ct subsets. 26.1.2. In what follows we always deal with so-called Borel probabdities which are defined on
a(5”).
A Borel probability ,u is said to be reguZur if p ( B ) = sup (,u(K):K E B, K E R(T)] whenever B @(T).A regular Borel probability is siniply called a Radon probubiZity.
25.1.3. The non-trivial direction of the following criterion is proved in [BAU, p. 2011. Proposition. A Borel probability defined OTLa complete metric y a c e .is regular if a d anly if i t .is macentrated on some separable subspace.
26.1.4. Obviously, every Radon probability ,u is uniquely determined by its restriction po to $(“). We now state an important characterization; [BOU3, chap. IX, pp. 42-47]. Proposition. A m a p po from 9(T)into &?+can be extended to a Radon probability if and o d y if the following conditions are satisfied: (1) Let H , hr E $ ( T ) such that 13 K . Then p o ( H ) po(K). (2) W e have po(K, u K,) 5 po(K,) ,uo(K,)for K,, K 2 E ft(T).Moreover, equcclity holds if K , and K , are disjoint. ( 3 ) Given KOE ji( T )and E > 0, there exists an open subset U containing KOsuch that p o ( R ) 5 p o ( K o ) E for all K E R(T)with K 5 U .
+
+
(4)sup { p o ( K ) :K E R(T)) = 1.
25. Radonifying Operators
349
Cylindrical Sets
25.2.
25.2.1. Let us recall that Cod ( E ) denotes the collection of all finite codiniensional subspaces N of the Banach space E. Given N,, N2 E Cod ( E ) such that N , N2, then x ( N l ) -+ x(N,) defines a canonical map Q from E / N l onto E{N2, and we have the commutative diagram
Hence, the family of all quotients EIN with N E Cod ( E ) constitutes a projective spectrum of finite dimensional Banach spaces. 25.2.2. If N E Cod (E),then &(E) denotes the a-algebra which is the inverse image of %(E/N) with respect to the canonical map Q:. Subsets of the form
2 := (x E E : Q ~ E xB ] , where B E B ( E / N ) , are called cylindrical. We put
8(E):= uN 3N@)* 25.2.3. Proposition. 8 ( E ) is an nlgebra. Proof. Let Z, a
...,2, E 8 ( E ) .
Then Zi E QNr(E), where Ni E Cod ( E ) . Since
N := n N i E Cod ( E ) ,it follows that 1
n
U Zi E 8n.(E) L 8 ( E ) . 1
The remaining properties of an algebra are trivial. Remark. Let us mention that 8 ( E ) is a a-algebra in the finite dimensional case only. 25.2.4. We now give some examples of cylindrical subsets.
Proposition. If K E R(E) and N E Cod ( E ) ,then K
+ N E S(E).
Proof. Since B :=-Q$(K) is compact, we have B E B ( E / N ) .Hence
K
+N =
(Z
E E: Q ~ EXB JE &v(E) S 3 ( E ) .
25.2.6. Proposition. A subset 2 of E is cylindrical if and only if it cnn be written in the form
z = :.( E E : ((5,a,>,...,(x,a,)) E 81, where a,,
...,a, E E' and B E
%(Sfl).
350
Part 5. Applications
25.3.
Cylindrical Probabilities
25.3.1. A map from 8 ( E )into 9+is called a cylindrical probability if its restriction to the a-algebra &(E) is a probability for all N C Cod ( E ) . 25.3.2. We now describe an important method to produce cylindrical probabilities. For this purpose an arbitrary probability apace (a. u ) and a linear map X from E' into Lo(S,p ) are required. Given any cylindrical subset = (Z E E : ((2,al},
we put
zx
= { w E Q : (f1(w),
...,(5,an}) E B } , ...,fn(u4) E B),
where f r := X a j . Since f l , ...,f n are actually equivalence classes, it should be mentioned that 2 , is well-defined up to a p-null set. So by setting p#) := p&) we get a map px from 8 ( E ) into 9+.
25.3.3. The following result is trivial. Proposition. px i s a cylindricctl probability. 25.3.4. For detailed proofs of the converse statements we refer to [BAD, exp. 31 and [SWA, pp. 256-2581.
Proposition. Every cylindrical probability c can be generated by some lirtear X from E' into Lo(O,p ) such that 5 = pX. Here (Q, p ) i s a suitable probability space. Proof (sketch). Let ( a i )with i E I be a Hamel basis of E'. If 0 denotes the I-th Cartesian power of X , then A x := ((2,aj))defines a map A from E into 8. For any finite index set i E g ( I ) we consider the canonical surjection Qi from 8 onto Q,, tlhei-th Cartesian power of X . Put Ai := QiA. Then the family of all images pi:=Aic with i E B(I) coi;stitutes a projective spectrum of probabilities. So, according to Kolmogorov's extension theorern, we can find a probability ,LC on D such that pl = Qlp for all i E S ( I ) ;cf. [RAU, pp. 157-1162] and [PEH, pp. 5-17]. Finally, the desired linear map X from E' into Lo@,p ) is given by I
iliai+x d i f i , Z
where f i denotes the i-th coordinate functionon 0. Obviously, if 2 = ( X E E : A i x E B ] , then ZX = ( w E Q: Q,co 23). Therefore pu,(B)= [ ( Z ) and pc,(B)= p(&) imply ( ( Z )= px(Z) for all Z 6 a ( E ) .This proves that 5' = px. 25.3.5. A cylindrical probability is called Badmian if, given K E B(E)such that 5(K N ) 2 1 - E for all N E Cod ( E ) .
+
E
> 0, there
exists
25.3.6. Proposition. Every cylindrical Radon probability t can be uniquely extended to a Radon probability f . Proof (sketch). In a first step we put
(,(R):= inf ( ( ( K+ N ) : N E Cod (E)]for K E L(E). Since tosatisfies the conditions stated iii 215.1.4,there exists a Radon probability f such that f ( K )= (,(K). Finally, we can show that 4 is tho required extension of C.
25. Radonifying Operators
Remark. Conversely, given any Radon probability, then its restriction to is of course a cylindrical Radon probability.
351
3(E)
26.3.7. We now give a fundamental characterization of cylindrical Radon probabilities which is the key to all what follows; of. [BAD, esp. 121 and [SWA, p. 2961.
Proposition. Let X be a linear map from E' into L,(Q, p). Then ,ux.is a cylindrical Radon probability i f and only i f there exists a p-measurable P-valued function x defined OTL Q such that
X a = @(.), a) for all a E E'.
In this case, the extension px equals the image x ( p ) . P r oof (sketch). Suppose that px is a cylindrical Radon probability. First we show that X becomes continuous if E' is equipped with the locally convex topology generated by the system of semi-norms
pK(a):= sup ( ) ( xa)): , x E K ) , where K E S?(E').
+
Tor this purpose, given E > 0, we choose K E R(E) such that pX(K N ) 2 1 - E wheneverN Cod (E).Takeanya E E'withpK(a) 5 Eandwritef := X a . Nowweput
N := ( x E E : ( 2 , a ) = 01, 2 := (xE E : I(%, a)l 2, := (a E Q: If(a)l
>E),
>E).
+
Then ( K N ) n 2 = 0 implies p(Zx)= px(Z)2 E . Hence IjXa)jo5 E. This proves the continuity of X . According to 26.1.3 we may assume that E is separable. Then there exists a countable subset (ai)of E dense in the locally convex topology defined above; of. [KOT, pp. 261 and 2661. Put f i := Xai. Let Xmdenote the countable Cartesian power of X . By setting A x := ((2,a,)) andf(0) := (fi(o)f we obtain a px-measurable linear map A from E into X" and a p-measurable X"-valued function f on Q. I n order to show that A($=) = f(p),the canonical surjection Q, from X" onto Xnis required. Put A , := QnA andf,, := Q,$f. If B b(X"),then we write
z:= (x€ E:A,x € B ) , z, := (0€ . R : f n ( 0 ) € B ) , 2, := {u E Y " :&,u
E B).
It follows from Z = A-l(Z0) and 2,
=f-l(Z0)
that
[ A ( i i x ) l W o= ~ i u X ( Z ) = cd(Zx)= [f(01)l(Zo)Since the Bore16-algebra of X mis generated by the subsets Z,, we have shown k k t A(,&) andf(p) coincide. Choosea sequence of subsets h', E a(#)with liin fiX(Km)= 1 ni and put,
Then
352
Part 5. Applications m
NOW,for every o 6 Qo, we select a(o)E A-l(f(o))n
u K,.
This yields an E-valued
1
function x defined almost everywhere on Q, Moreover, ( ( ~ ( w )n,i ) ) = A ( z ( o ) )= f(o)>= ( f i ( ( o ) )
wlienevor
GO
E Q,,
irnplies Xui = (a(.), ai) for i = 1,4, ...Using the c0ntinuit.y of X we get X a = ( ~ ( . ) , n ) for all a. E E’. Since the scalar functions 112 we get the diagram €2
,Lp[O, 11
ClO, 11 JP
The assertion now follows froin 17.3.5.
26.6.4. The Brownian ?notion can be described by the so-called cylindricul Wiener defined on Co[O, 11 := { f C[O, 11: f ( 0 ) = 0). probabilit?yg := R(yLZIO,ll)
28.6.5. Immediately from 25.6.3, 25.4.8 and 25.5.4 we have the Theorem. The cylindrical Wiener probability e can be uniquely extended to the Wiener probability @ which .is regular on the Bore1 a-algebra B(C,[O, 11).
26.6.6. Finally we show that the Wiener probability is concentrated on certain linear subsets of C,,[O, I]. Theorem. If 0
< i, < 1/2, then P(CJ0, 11) = 1.
Proof. Choose some exponent p such that 112 - A > 1/21 > 0. Then we have R E ?&,(,(LBIO, 11, C1[O, 11) by 25.6.3. Hence el := R l ( y L g [ O is , l la) cylindrical Radon pro-
bability on C#I, 11. Moreover, @ is the image of el with respect to t.he embedding map from C2[0,11 into Co[O,11. This proves the assertion. Remark. The above result means that almost every sainple path is Holder A-continuous €or 0 < 1 < 112. Remark. Let us mention that C1[O,13 with 1/2 < i( < 1 is a @null set; cf. [KUO, p. 451 and [YEH, p. 4361.
25. Radonifying Operators
36.7.
357
Notes
Cylindrical probabilities on special linear topological Hausdorff spaces were first studied by -1.N. KOLWOQOROV in his fundamental monograph [KOL]. Further results were obtained by several Soviet mathematicians; cf. [GEL]. Finally, L. SCHWAETZ [l], [2], [3] introduced the concept of a radonifying operator. This famous theory is presented in [SERI,, exp. 1--6, 11-17, 24-26]. See also [BAD, exp. 121. The reader should also consult the monograph [SWA]. It wns S. K W A P I E[3], ~ [4], 151 who discovered the coincidence of absolutely p-summing and p-radonifying operators for 1 < p < cy?. There is an extensive literature dealing with Gaussian probabilities on Banach spaces; cf. [BAH], [KUO], [YEH]. Recommendutions for further reading:
[RAU], [BOU,, chap. 1x1, [HAL], [SHI], [SKO], [XIA]. 1’. ASSOUAD 113, G. BENNETTCS], R.. M. DUDLEY [l], D. J. H. GARLIXG[4J, [7], W. LINDE [ i j . [4]$W. LIXDE/A.PIETSCH [l], B. YAUREY [2], N. WIENER [l].
26.
RierJz Theory
The background of this chapter is the classical theory of compact operators created by I?. RIESZ.Looking for significant properties of I - 23,where 8 is compact, A. F. RUSTON axiomatically introduced the class of so-called Riesz operators in 1954. Our main purpose is to investigate operator ideals U such that all components
U(E,E)consist of Riesz operators only. The largest operator ideal of this kind was defined by D. KLEINECKE. Further examples are the ideals of compact, strictly singular, and strictly cosingular operators. It is essential t,o emphasize the great influence of the famous Soviet School on the development of this theory. First we recall the Riesz decomposition of operators possessing finite ascent and finite descent. Secondly, some elementary results concerning spectral theory in Banach algebras are summarized. Then we introduce the basic concept of a @-isomorphism due to F.V.ATKINSON(1951). The next step is the theory of quasicompact operators developed by K. YOSIDA(1939). Using these preliminaries we define Riesz operators and establish their main properties. The last section is devoted to ideals of Riesz operators. Throughout this chapter all Banach spaces under consideration are complex.
26.1.
Riesz Decomposition
26.1.1. I n this section some purely algebraic results are collected; cf. [TAY, p. 2711. We always consider a linear map T acting in a linear space E. 26.1.2. The map T has finite ascent if there exists an integer k 2 0 such that N ( T k )= N(Tk+l),The smallest such integer is denoted by n N ( T ) .Then we have N(Th)= N(Tk)for h > 12 2 nN(T). 26.1.3. The map T has finite descent if there exists an integer k 2 0 such that M(Tk)= L!kf(l'k+l). The smallest such integer is denoted by nnr(T).Then we have N ( ! P )= M(Tk) for h > k 2 niM(T). 26.1.4. Proposition. Let T have finite ascent and finite rle-scent. Then na(T) ad naf(T)are equal. Remark. The common value of nLV(T) and ,z,(T) is denoted by a(T). 26.1.5. We now describe the so-called Riax decomposition. Theorem. Let T have finite ascent and finite descent. il'hen E is the direct sum of N ( P ) and M ( T k )for k 2 n(T).
26. Riosz Theorv
26.2.
369
Spectral Theory in Banaoh Algebras
26.2.1. In this section let 9f be a complex Banach algebra with identity I ; of. [HILJ. 86.2.2. For S E 9f we put
~ ( 8:= ) (1E V : I The element #(A)
- 23
:= S(I -
is invertible).
is called the Fredholm resolvent.
26.2.3. An easy computation proves the Lemma. For all il E e(S) we have ( I - 1~9)-~ =I
+ AS(l).
86.2.4. Theorem. T?he function S(.) i s analytic on the open set e(8). 26.2.6. Theorem. The condii?i~ns lim IlS"l[l/"= 0 und ~ ( 8 =)V are equivalent. n
26.3.
@-Isomorphisms
26.3.1. An operator T E B(E,F ) is called a @-isomorphismif it has a finite climensional null space and a finite codimensional closed range. R e mark. Usually @-isomorphismsare called Freclholm operators or Noether operators; cf. [SEC, p. 1061 and [PRO, p. 161. Remark. Let us mention that H ( T )is automatically closed if it has finite codimension; cf. [CAR, p. 371.
26.3.2. We now prove a fundamental characterization. Theorem. An operator T E B(E,F ) i s a @-isomorphism i f and only i f there exist operators U , B E B(F, E ) , X E g ( E ,E ) , und Y E $ ( F , P)such that
UT
=
IE - X
and
TV
= I, -
Y.
Proof. Let N := N ( T )and 2M := I M ( T ) . By A.3.4 everyd5-isoniorphismT E B(E,F) admits a factorization T =; JToQ, where To is an isomorphism. Moreover, it follows from B.4.4 and B.4.5 that there are X, E B(E/N, E ) and YoE B(F, M ) with QX, =IE,N and Y,J = IM. If
X := Ie - XoQ, Y := IF - J Y , , and 5' = V := .XoT;'Yo, then
UT
==
XoT,'(YoJ) T0Q = XoQ = I,
-
S
-
Y.
a d
T B = JTo(QXo)T i l Y o = J Y ,
=I,
iV and M(Y)= M , the operators X and Y are finite. Conversely, let us suppose that UT = I E - X a.i;d T V = IF - Y , i%-hcre X and Y are finite. TheE N ( T ) 5 N(IE - X ) . Hence Y ( T )E Dim ( E ) .The range of T is closed, since X(T ) = X ( I F - Y)+ Mo with a Gnite dimensional subspace Me. T h i s proves Since X(X)
7
that M ( T ) E Cod (3').
360
Part 5. Applications
26.3.3. The next statement is the main result of the classical Riesz theory. T h e o r e m . For 8 E R ( E , E ) the operator I - S is a @-isomorphism. R e m a r k . If we consider operators acting in a fixed Banach space E , then the identity n a p I E is simply denoted by I.
26.3.4. Let Q, bc the quotient map from B(E, E ) onto the so-called Calkin aEgebra
O(E, E ) / R ( E ,E ) . Then we have the T h e o r e m . An operator T E B(E, E ) i s a @-.isomorphi.m if a d only if @(T)is iiwertible in B(E, E ) / R ( E ,E ) . P r o o f . The formulas UT = I - X and T P = I - P stated in 26.3.2 yield that @ ( T )admits a left-hand inverse and a right-hand inverse in B(E,E)/JI(E,E). Conversely, if @(T)is invertiblc in B(E,E ) / R ( E ,E ) , then UT = I - X and TV = I - Y, where T:, V E P(E, E ) and X, Y E R ( E ,E). By 26.3.3 we can find Uo,Vo E B(E, E ) and X,,Yo E %(E,E ) such that Uo(I - X ) = I - X , and ( I - Y ) Vo = I - Yo. Hence U0UT = I - Xo and T V V , = I - Yo. This completes the proof.
36.4.
Quasi-Compact Operators
26.4.1. An operator S O(E,E) is called quasi-compact if there exist a n operator K E R ( E , E ) and a natural number m such that llSm - Kjj < 1. 26.4.2. Lemma. For every qwtsi-compact operator S f B(E, E ) and E > 0 there exist an operator L E R(E,E ) and a natural number 9% such that \ISn- LII < E .
Proof. Choose K E R(E, E ) and m such that llSm- KII < 1. Let us define the operators Kh := Shm- (Sm- K ) h for h = 1, 2, , Then K h E R(E, E ) . Moreover, llShm- R h l l 5 \ISm- Kllh < E for h large enough.
..
26.4.3. P r o p o s i t i o n . Let S E O(E,E ) be quasi-compact. Then I - S is a @-isomorhism.
Proof. We choose K E R ( E , E ) and m with IISm- KI] < 1. If A := Sm- K , then I - A is invertible. Consequently,
(I - A)-l(I + s+ * * *
+ s-1) (I-
As)
= I - (I - A ) - l K
and
(I - S)(I f S
+ - + 8-1) ( I - A)-1 = I - K ( I - A)-1. **
Finally, the assertion follows from 26.3.4.
26.4.4. P r o p o s i t i o n . Let S O(E, E ) be quasi-compact. Then I ascent.
-
S has finite
Proof. Suppose that the sequence of subspaces N ( ( I - 8)k) is strictly increasing. Then, by E.l.l, there are xk E X ( ( I - S)k)such that l l q l l = 1 and llzk - 211 2 1/2 for all 2 E N ( ( 1 - A’)k-l). We now choose L E R(E, E ) and n with /IS” < 1,’s.
26. Riesz Theory
361
If xi := xk - S n q , then
x i = (I
8
+ + sn-l)(1 - 8 )xk E N ( ( I - s)k-l)
and
Lxk
xk - x! - (fin- L ) 3 .
Consequently, for h
> k we have
[ILxh - Lxkl] 2 11xk - (x:
+ xk
-
xi)ll - 2 118' - LII
> 1/6.
Therefore (Lxk) can contain no convergent subsequence, and this contradicts the fact that L is compact.
26.4.6. Proposition. Lef S
< B(E, E )
be quasi-compact. Then I - 8 has finit
dcscent.
Proof. Suppose that the sequence of subspaces M ( ( 1 - 8)k) is strictly decreasing. Then, by E.1.1, there are %k E M ( ( I - S ) k )such that llxkll = 1 and llzk - 211 2 1/2 for all x E H ( ( I - S)k+l).We now choose L E R(E,E ) and n with 11s" - L11 < ll6. If x i := xk - Saxk, then
xi
=(I
+ s+ + * *
( I - S ) Xk E M ( ( 1--- S)k+l)
s 1 )
and
Lxk = xk - x i - (8" - L) 9 . Consequently, for h j/L%k
< k we have
- Lxkll
2 1121 - (x! - xk
f
%!!I1 - 2 118' -
> 1/6-
Therefore (Lxk) can contain no Convergent subsequence, and this contradicts t h e fact that L is compact.
Riesz Operators
26.6.
26.5.1. An operator S E B(E, E ) is called a Riesz operator if I - ji,Sis a @-isomorphism for all complex numbers A. 26.5.2. First we give the basic examples; cf. 26.3.3. Theorem. Every S E R(E, E ) is a Riesz operator.
26.5.3. Lemma. Let S E O(E,E ) be a Riesz operator. Then A S is quasi-conzpmA for rcll complex numbers A. Proof. According to 26.3.4 we have Q(@(S)) = V. Hence, by 26.2.5, it follows thct lim ~ ~ ~ @ ( S= ) n0, ~where / ~ l ' n111.111 is the norm of thecalkin algebra B(E,E)/&(E,E). U
Since, given 2. E %?, there is it natural number m such that //l@(itS)mllil'm c 1, we can choose K E R(E, E ) with Il(A8)" - K(I < 1.
26.5.4. A coinpiex number & is called a characteristic value of the Riesz operatm S - AoS)-f (01. We put n(Ao)I== n(1 - A ,&').
if N ( I
I
Remark. Obviously A, is a characteristic value if and only if 12,'
is an eigenvalue
362
Part 5. Applications
26.5.5. Theorem. For every Riesx operator S E E(E,E ) the set of characteristic values has no finite point of accumulation. Proof. Suppose that 8 has a convergent sequence ( 1 , ) of distinct characteristic o with x k = 1fiXk:lr. Since x l , .., xk are linearly indevalues. Then there exists x k pendent, the sequence of subspaces Nk spanned by { x l , ...,xk} is strictly increasing. Consequently, by E . l . l , there are yk E Nk such that l\1/kll = 1 and llyk - yll 2 1/2 for y E Nk-1. We now choose L E B(E, E ) and n with Il(nS)n- L(I < 116, where A := lim Then there exists ko such that Il(&s)" - L11 < 1/6 for k 2 ko. Since
.
+
k
(I - (&&)") xh = (1 - (&/&)m) x h for h = 1, ...,n , the element y: := 1/k - ( j l k S ) n 1/k belongs to Nk-1. Moreover, Lyk = Y k - ?-/: - (&@" - L,1/kFor R > k 2 ko it follows that
- Lykl! 2 lk/h - (d - 1/k f
lb%h
d)ll - ll(hs)"- LII - l (&s)m
- LII >
Thus (@k) can contain no convergent subsequence, and this contradicts the fact that L is compact. R e m a r k . The set of characteristic values is a t most countable.
26.6.6. T h e o r e m . If I i s not a characteristic value of the R i m operator 15' E 2(E,E ) , then 1 E e(S). Proof. Since N ( I - 1.S) = lo), we have n N ( I - AS) = 0. Consequently, nirr(I- I S ) = 0 by 26.1.4. This means that M ( I - AS) = E . Therefore ( I exists. 26.5.7. As a summary of the results given in 26.1 and 26.4 me formulate the Theorern. If A. is a characteristic value of the Riesz operator S E f?(E,E ) , then, setting p := a(,lo),the following properties hold: (1) The subspaces N ( ( I - ?.oS)k) are finite dimensional and (0)
... c N ( ( I
-
1.~8)p-i)
c N((I-
a,,s)p) = N ( ( I - a o s ) p + l ) = ... .
(2) The subspaces M ( ( I - l o S ) k )are finite codimensional and
M ( ( I - 1oS)P-l) 2 M ( ( I - Il.oS)P) = N ( ( I - A$)P+') Furthermore, E is the direct sicm of W ( ( I - IoS)P)and M ( ( I - AoS)P).
E
3 *-. 3
=
.
36.5.8. Theorem. If lo ,is a characteristic value of the Riesz operator S E B(E,E), then there exists a decomposition 8 = 8, -k Snfwith the follounhg properties: (1) il, i s the only character&tic aalue of Sa E g(E, E ) , (2) I,o is not a characteristic value of Sll f g ( E , X ) , (3) S-VSJf = S,S, = 0. Proof. We represent E as the direct sum of the subspaces N := N ( ( I - AoS)p) and M := X ( ( I - 4,S)p). Let PN and PMbe the projections from E onto N along M and from E onto M along N , respectively. Put S, := PNSPNand SM:= P M ~ P , . Since N and M are invariant under S, we have Sx = Savxfor x E N and Sx = SMx for x E M .
26. Riesz Theory
363
(1) Suppose that I f I , is a Characteristic value of 8,. Then there exists x f o with (I - AS,) x = 0. Hence x E 37 and Sx = S,X. Since ( I - ?&')p x = o and x = B x , it follows that (1 - Ao/?,)p x = 0. This implies x = 0, which is a contradiction. (2) Suppose that, & is a characteristic value of S., Then there exists xo $. o with ( I - A&,) zo= 0. Hence xo E M and Sxo = SMxW Now it follows from ( I - A,#) xo = o that xo E N . Since N n M = ( o } , we obtain xo = 0, which is impossible. (3) The assertion follows from PNPM= P B ~ P= N 0.
26.6.
@-Injectionsand @-Surjections
26.6.1. An operator T E e ( E , F ) is called a @-injection if it has a finite dimensional null space and a closed range. 26.6.2. Obviously we have the following criterion. Proposition. An operator T E B(E, F ) is a @-injection if and only if there d s t s a finite codirnewionalsubqwm. M of E amh that T J S is an injection.
26.6.3. We recall that 6 denotes the ideal of Kato operators. Theorem. Let S E G(E, P ) .If T E 2(E, F ) 13a @-injection,then so is S $- T . Proof. By 26.6.2 we can find M E Cod ( E ) such that TJ& is an injection. Suppose that S T is not a @-injection.Then, by 1.9.1, there is an infinite dimensional subspaoe M , of M with ]l(S T )J&]I < j(TJ:) 5 j(TJ&). Now it follows from
+
+
i(sJ&o) 2 j(TJ$J - Il(S + T)J%,11 > 0 that SJ& is an injection. Therefore &lo must be finite dimensional, which is a contradiction.
26.6.4. Lemma. Let T E B(E,F ) be an injection. If A E 2 ( E , F ) and lIAl1 < j ( T ) , then T A .is an injection. Moreover, N ( T )E Cod ( F ) and M ( T A ) E Cod ( F ) are equivalent.
+
+
Proof. By 13.3.11 we have
+ LA)2 j ( T )- iiAll whenever 0 5 I In particular, T + A is an injection. j(T
4 1.
Suppose that one and only one of the subspaces M ( T )and X(1' codimension. Then, given E > 0, we can find lo,ill E [0, 13 such that
:Ifo := X ( T and (Io -
< F.
+ I.&)
E Cod (Fj. M I := M ( T
+ AlA)
By E.l.l there are yI,y2, ... E UFwith
+ A ) has finite
Cod ( F ) ,
364
Part 5. Applications
On the other hand, sime F / M o is finite dimensional, we have flyi(Mo)- yk(Mo)li< E for some i =+ k.
c'hoose x E E such that ( j V )-
lIzji - ?Jk -
IIAIl) l!.~!l5 j(T , 2 ljzji
(T + &A) xII < E . Then it follows froin
+ AoA)1/415 iU' -+ AoA)4 -
ykll
-+
E
52 +E
that
which becoiiies false for
E
small enough. This coinpletes the proof.
26.6.5. Theorem. Every S E G(E,E ) is a Riesz opercrtor. Proof. We see froin 26.6.3 that I - ils is a @-injection for all complex numbers A. Given & E V , by 26.6.2 we can find M E Cod ( E ) such that ( I - A,#) J$ is a n injection. Let 11. - ilol 1 1 8 1 1< j ( ( I - A0S)J:l). Then, by the preceding lemma, the subspaces N((1- &S) J:) and M ( ( 1 - ils) are finite (infinite) dimensionaJ simultaneously. This statement remains true for M(1 - &S) and M ( I - AS). Therefore, the complex plane is the union of the disjoint open subsets
JC)
Go := ( 7 6 E %: M ( I - 2 8 ) E Cod ( F ) } and GI
:= (3. E %':
M(I - 2 8 )
6 Cod (3')).
Since 0 E Go, i t follows t,hnt GI= 0. Consequently I - ih' is a @-isomorphismfor all complex numbers A.
96.6.6. An opemtor T E 2(E,F ) is called a @-surjectiow,if it has a finite codimensiond closed range. 96.6.7. Obviously we have the following criterion. (I
P r o p o s i t i o n . An operator T E B(E, F ) i s a @-surjection if and only i f there &ts finite dimensional subspnce N of E such that Q:.T is a surjectioit.
96.6.8. We recall that 2 denotes the ideal of Pekzyiiski operators. Using the method of 26.6.3 we get the T h e o r e m . Let 8 E Z ( E , F ) . If T E B(E, F ) i s a @-swjection, the72 so is S
+ T.
26.6.9. The ncxt result follows from 26.6.4 by dualization; cf. B.3.8. Lemma. Let T 2 O(E,P) be a surjection. If A f e ( E ,F ) and IlAII < q(T), the)% T A i s a suvjection. Noreover, N(T)E Din1 ( E ) and N(T A ) E Din1 ( E ) are equivalent.
+
+
20. Riesz Theory
866
26.6.10. Analogously to 26.6.5 we have the Theorem. Every S E z ( E , F ) is a Biesz operator. €3) be the set of all operators S E E(E, E ) such that the pertur26.6.11. Let bation S + T of any @-injection T B(E, E ) is also a @-injection. Analogously Z,(E, E’) is defined to be the perturbation set of @-surjections.
26.6.12. A proof of the following result is given by [CAR, p. 971. Theorem. G,(E, E ) und Z,(E’, E ) are c l a d ideals in the nlqebru B(E,E ) . Remark. It is unknown whether the components Ga(E, E ) and &(E, E ) satisfy the compatibility condition stated in 1.1.3. The answer to this problem is affirmative if and only if Ga(E, E ) = G(E,$2) and Z@(E,E ) = Z(E, E ) for all Banach spaces E .
26.7.
Ideals of Riesz Operators
26.7.1. First of all let us inmtion that, in general, the set of Riesz operators is not an ideal in the algebra B(E, B). To give a Counterexample we consider the Cartesian square E x E of any infinite dimensional Banach space. Put Sl(zl)z2):= (0,x,) and S&, x2) := (z2,0). Obviously S, and S2are nilpotent. Hence they must be Riesz operators. On the other hand, S, S2 and XIS2cannot be Rieszian, since the characteristic value 12, = 1 has infinite algebraic multiplicity.
+
denotes the ideal of Gohberg operators; uf. 4.9.5. By 26.7.2. Let us recall that 4.3.8 and 26.3.2 we have the Theorem. R .is the largest wpercrttw d e a l 8wh that all co?nponen,ztsR(E,E) COPZs&t of Riesz operators only.
26.7.3. In the following diagram thc arrows point from the smaller operator ideals at the larger ones:
R
All inclusions are strict.
366
Part 5. Applications
26.8.
Notes
As everybody knows the famous spectral theory of compact operators was created by F. RIESZ [l]. Further contributions are due to many authors. I n particular, @-isomorphisms were investigated by F. V. ATKINSON [l] and I. C. GOHBERO/N.G. KREJN[I]. The concept of a quasi-compact operator was found by K. Yosma 111, and A. F. RUSTON[4] introduced the so-called Riesz operators. The largest ideal consisting of Riesz operators first appeared in a paper of D. KLEINECKE[I]. See also I. C. GOHBERG/A. S. XARKUS/I. A. FELDMAN [l] and B. YOOD [I]. Remmmendations for further reading: [CAR], [DUN], [GOL], [HEU], [KAT], [PEO], [RIE], [ROL], [SEC], [TAY], [ZAN].
s. R. CARADUS[I], D. HIL3ERT [l], T. KATO[I], S. N. GAEHOWSKIJ/A. 8. DIKAESKIJ [I], A. PIETSCE [I], A. F. RFSTON[5], &I. SCHECHTER [l], E. SCHMIDT [l], [Z].
27.
Fredholm Theory
The famous Fredholm theory of integral operators with continuous kernel was the most important starting-point of functional analysis. All determinant-free results have been generalized to the Riesz theory developed in the preceding chapter. On the other hand, for a long time no one succeeded in constructing determinants for operators in arbitrary Bansch spaces. This problem was independently solved by A. F. RUSTON, A. GBOTHENDIECK,and T. L E ~ A ~ ~inS the K I early fifties. We begin with a fundamental characterization of Riesz operators in terms of the meromorphic behaviour of their Fredholm resolvent. By Weierstrass's theorem, there exists an entire complex function the zeros of which are the characteristic values of the given operator. Then the Fredholm resolvent is the quotient of an entire operator-valued function and the so-called Fredholin divisor just described. The main purpose of this chapter is a direct construction of Fredholm divisors. Models are the characteristic polynomials of matrices, H. v. KOCH'Sdefinition of infinite determinants (1900)' and the above mentioned theory of integral operators. As a first step we form Fredholrn divisors for nuclear operators in ZI. Using the ooncept of related operators the results can be carried over to some classes of operators acting in arbitrary Banach spaces. I n particular, we obtain the Ruston-Grothendieck determinant theory of nuclear operators. It was proved by I. S c m (1909) that for every Hilbert-Schmidt operator the sequence of reciprocal characteristic values is square summable. A general theorem of this kind, concerning Gp-operators with 0 < p c 00, is due to H. WEYL(1949). Again using the concept of related operators we study the distribution of characteristic values for both (r,1, 2)-nuclear and absolutely 2-summing operators. Finally, a trace formula is established which goes back to A.CROTHENDIECK (1955) and V. B. LIDSKIJ(1959). Throughout this chapter all Banach spaces under consideration are complex.
27.1.
Fredholm Divisors
27.1.1. The main result of this section is the following characterization of Riesz operators in terms of the meroniorphic behaviour of t,heir Fredholm resolvent. Theorem. An operator IS E B(E,E ) is a Riesz operator if and o d y if for every complex number lothere exists e > 0 such that
for 0 < /A - AOl n = 0, 1,
...
< e,
where IS-p(lbo), ...,IS+@,) E g ( E , E ) and Sn(Ao)E B(E,23) for
Proof. Let IS be a Ricsz operator. If Jo E e(S),then by 26.2.4 the function S(.)hae a Taylor expansion at A,. We now consider a characteristic value A,. Then thers
368
Part 5 . Applications
+
exists a decomposition S = BN SM according to 26.5.8. The finite operator SAv has an g ( E, ,!?)-valuedFredholm resolvent
+
for i, A0; cf. [ZAN, p. 3461. On the other hand, SM(.)is analytic in a certain neighbourhood of &. It follows from SNSM= SMSN= 0 that S(A) = s&) s,().). This gives the desired Laurent expansion.
+
Conversely, let us suppose that the above condition is satisfied. Then the complement of e(S) has no finite point of accumulation. Since the coefficients of all principal parts are finite operators, the function S(.) is analytic modulo R(E, E ) in the whole coniplex plane. Hence, S is a Riesz operator by 26.3.4. Remark. The operator SAV can be obtained from the Laurent expansioii of S(A) at &, since S,(A) coincides with the principal part and 8, = S,(O). '37.1.2. We now state an improvement of the previous theorem.
Proposition. Let S E 2(E,E ) be a Riesz operator. Then tfhe Fredholrn re.sdvent hns CI pok of order n(1,) for every characteristic value lo. Proof. Let
for 0
< /;I- jlol < e
such that S-,(L,)
+ 0. Then lim (1 - 1,). s(2)5 = o for all
x 5 E m d n > p . Wenowsuppose that 2 E N ( ( I (I -
z = ((a
- a),
ri-*lo
It follows from
s +- (I - ns)p x
by niultiplication with ( I - AS)-1that
+ (I - A s ) n - l z
= 0.
i E 1. tends to lo we obtain ( I - l,,S)n-l z = 0. This proves that we have ~ ( (-1$S).-1) = N ( ( I - i.,S)n) for n > p . Consequently, ~(1,)5 p .
Conversely, ( I - H ) S ( i ) = 8 iinplies ( I - 1,S) S(1) - (A - 2,) SS(1) -::S. Comparing coefficients we see that
(I - /'.,A') S-,(i,,)
= SK,(L0),
( I - 1,s)As-p+l(ao) = xs-,(a,). ( I - RoS) S&))
= 0.
27. Fredliolm Theory
The last equation means that &,(Ao) (I - M)*1
= (I
369
= A,,AS’LS-~(~.~). On the other hand, we have
- jl,,AS’)P-Z SS-.z(&)= - - -
= SP-lS-,(&).
Consequently, ( I - &!3)P--1
K1(&) = A;p+lS&,)
Choose 2, E E with &-,(lo) so
+
0.
S-l(2.0)xo E “((I - @)P)
and ( I - 2&)P fLl(&) = 0.
Then \ N ( ( I - l.os)p-l).
Hence p I n(Ao).
27.1.3. For every characteristic value A,, of a Riesz operator S E E(E, E ) the nlgebraic multiplicity is defined by := dim N ( ( 1 - @)p),
where p := ...(lo).
5 a(&). Proof. Choose 2 E N ( ( I - A&)”) \ N ( ( I - 2J3)P-l) and put := ( I - &8)k-1 z for k = 1, ...,p . Then sl,..., zpare linearly independent and belong to N ( ( I - &9)p). 27.1.4. Lemma. n(&)
27.1.5. An entire complex function d is called Fredholm divisor of the Riesz operator S if the zeros of d coincide with the characteristic values of S. The order of every zero being the same as the algebraic multiplicity of the corresponding characteristic value. 27.1.6. By Weierstrsss’s theorem [TIT, p. 2461 and 26.5.5 we have the P r o p o s i t io n. Every Riesz operator possesses cc Predholm divisor. 27.1.7. Finally, a very important result is shown.
Theorem. Let S E f!(E, E ) be a Rieez operator with a Fredhdm divisor 00
d(l) =2 6,P
for all 1 E 27.
0
Then there czists an entire E(E, E)-valued funci?ion m
D(i.1 = 2 D,A”
for all ilE %?
0
such that
Moreover, Do = 6,s and D, = 6,S f D,-,S for
n =
1,2,
...
Proof. Obviously D(A) := d ( l ) B(L) is an analytic function on e(S).By 27.1.2 and 27.1.4 all characteristic values of S are removable singularities of D. Consequently, a
24 Pietsch. Operator
3'70
Pnrt 5. Applicat,iorls
there exists an analytic extension to the whole complex plane. Comparing coefficient.;. from D(3.) (I- 1.8) = d ( i ) S we obtain the desired recurrence forrriuh. Remark. Let [2t, A] b e
EL
qiimsi-normed operator ideal. If the Riesz opzrator
S E 2 ( E , E ) bt-lo~gsto 3 , then so c!oc.s D, for n = 0, 1, . . . and 17eIVei ?n. This proves that the Fredholni detcririinant of S coincides with the characteristic polynomial of the ( n z , nz)-matrix ((q, aJ).
27.2.16. In the sequel we have to distinguish between different operators. For this purpose, if S E %(Z1, ZI), the Fredholm determinant of S and thcir coefficients are denoted by d(A, S) and an(&’). respectively. SfiIc %(11, ll). Furthermm, suppose that S P r o p o s i t i o n . Let S,SN, and = SfiISN= 0. T h e n d(A, S ) = a(?&, 8,) d(A, Sip,).
= flN
+ S,v1
Proof. The following formula will be shown by induction:
Obviously 6,(S) = 6,(S,) B,(S,). Kow suppose that (*) is correct for n trace = trace (SF-’) trace (Sg--“) we obtain 1 rn-1 & ( S ) = -&(S)trace ( t P k ) nz k=o
+
z
I
m-1
k
1
m-I
k
< m. Since
27. Fredholm Theory
375
27.2.17. Finally, we check the fundamental Theorem. For e w r y operator S E !&(Z1, diviwr.
I,) the Fredholm determinant .iS a Fredliolwi
Proof. The characteristic values of S and the zeros of d(3,, S)coincide by 27.6.11 and 27.2.12. Let a'(&, S) = 0 and use the decomposition S = SA,+- 8, described in 86.5.8. m
.
Consider some representation SN = 2 ai 0 xi such that (q,.., x,) is n basis of 1
X := ?$((I - nos)"), where p := ~ ( 2 ~Suppose ) . that det ((xk, ai)) = 0. Then there m
m
exists a linear combination x :=
?@k 1
+ 0 with 2
?.k(xk.
ai) = 0 for i = 1. ...,m.
k=l
Consequently, Sx = Ssz = 0. Therefore, it follows from ( I - & 5 ) p x = o that s = o, which is a contradiction. This proves that det ((xk, a i ) ) =# 0. By 27.2.15 we know that the Fredholm determinant of S , is a polynoinial of degree nz = a(&). Since Izo is the only characteristic value of XAv, we have
d(a, s,)
==
(1
-~i;i~)m.
On the otber hand, 26.5.8 (2) and 27.2.12 imply d(Jbo.S,w)=/= 0. Finally, the factorization d(2,S) = d ( l , 5,) d(il, S,) shows that lois a zero of d ( l , S) wvit,h order nz.
37.3.
Related Operators
2i.3.1. Operators:& E B(E, E)and T E B ( F , F ) are related if there cxist:d m d B r B(F, E ) such that S = R-4 and T = A B :
E
B(E, 3')
5.3.2. Related operators have many properties in cominoii. P r o p o s i t i o n . Let S E
B(E,E ) and T E B(F,F ) be related. Then p(8)
P r o o f . An easT; ccjniputation shows that S(2) = S and T(3.)= T US'(%) B for all ii E ~ ( 8 ) .
+
+- IBT(Z.)A
=~(1).
for all 7. E ?(T)
25.3.3. P r o p o s i t i o n . Everyoperutor SE B(E, 14;) relatad toa Riesz operntor T E 2 ( E , F ) dw 7 Biesz operator. Moreover, S and T hnec the some characteristic values with the s m i e crlrjebrnic niultiplicities. @ Proof. Suppose that S = BA and T = BB, where ,4 E B(E, F ) and €3 E 2(F,E ) . Sinoa 8 is a Riesz operator, given 3. E V, wc can find U 5 2 ( E , E:) and X E g ( E , E ) 1'9
376
Part 5. Applications
with U(IE - ils)= IE - X . Put Uo := I F + AT
+ PAUBT, then
UO(IF- RT) = IF - 12T2 + A2AUBT(I, - AT)
+ A2AU(IE As)BT = I F - i2T2 + A’A(IE X ) BT
= I F - li2Ta
-
-
=;-
I F - A’AXBT.
Hence Uo(IF - LT)= IF - X , wit,hX o := IZAXBT. Analogously we may construct Vo E e ( F , F) and Yo E g(P,P ) such that (IF- AT) JT, = IF - Yo. So IF - I T is a @-isomorphism.Obviously A induces a one-to-one map from N((I E- Uj”)onto
N((IF- AT)n) for
18
=
1,2, ... with inverse jJ3
(-AT)”l. In particular,
k=l
it follows that) dim ( N ( ( I ,- AS).)) = dim ( N((I F- AT);)). 27.3.4. As an immediate conscquerice of 27.3.3 we obtain the
T
Proposition. If the Riesz operator S E B(E, E ) is related to the Riesz operaior O(P,F ) , then every Predholm divisor of T is also a Fredholm divisor of S.
27.3.6. Now some important examples of related operators are considered. Proposition. Every operator S E %(E, E ) G re&d to an operator T E %(Zl, Zl). eo
Proof. Let S = 1ciaj $j)xi be any nuclear representation such that llajil and llxill = 1. Put
=
1
ca
-4%:= ((z,
ai)), sO(‘$j) :=
(ojtj), x(fk):=
fkxk. 1
Then S admits the factorization R
x
A 4
as described in 6.3.3. Observe that S = XSoA and T := SOAXare related. Moreover, T is a nuclear operator in 1, with the representing matrix (ci(rk, ai)).
27.3.6. By 27.3.4 and 27.3.5 we can carry over the results of Sect,ioii 97.2 to nuclear operators in arbitrary Banach spaces. ca
Theorem. Let S E R ( E , E ) . Por any nudear represet~ationS = 2 opj @ xi put 1 So := 1 and
27. Fredholm Theory
377
Then W
d(1) :=
2' 6 , P
for all 1 E V
0
is a Fredholm divisor of S .
Remark. The Fredholm divisor obtained in this way depends on the specid choice of the nuclear representation. This fact is a coiisequence of the negative answer to the approximation problem.
27.3.7. We may ask whether the determinant theory of nuclear operators can be T E %(C[O, 11, derived from the classical Fredholm theory. P r o b l e m . Is every operator S E x ( E , E) related to some integral operator T f a(C[O, 13, C[O, 11) admitting a representation by a continuous kernel?
37.3.8. Proposition. Every operator S f !4,,1,2,(E,E ) , where 0 < r 2 2, is related to an operator T f G&, h). Moreover, given E > 0, the operator T may be chosen such that 5 (1 E ) N ( r , l , z ) ( 4 -
+
W
Proof. Let S -=
cia9@ xi be 1
uj
2 0, Mu9) 5 (1
+
8
(r, 1, S)-nuclear representation with
8) N(r,2,1)(8), w2(ai) =
1,
ww(xk) =
1.
If W
~4% := ((2,a*)), o(li):= (C;-"'[j),
x(ik)
:=
lkc'$2q, 1
then we obtain the factorization S
E
+E
Observe that 8 = X D A and T := DAX are related. Since A X is represented by the matrix (C;!'(Z~, ai)),it follows that GL
Furthermore,
I(Xk, C&i)l5 zy
Ci}.
373 -
Part 5. Applications
Romark. We refer to [DUX, p. 11061, [GOH, pp. 1%- 1711, a n d I.RIN, pp. 109 to 1411 for a determinant theory of G;,-operatorsin I,. The Bretllzolin divisors obtaint7il thcw can be carried over to ( r , 1, 2)-nuclear operators by 27.3.4.
27.3.9. Proposition. E u e y operator S E ?&(E', E ) i.s related to a Hilbert-Schmidt operator T E G2(L2(K,y ) , L,(K, ,a)), where ,u is a probability on a suitable conipnct Hqusdorff space K. Moreover. the operator T may be chosen such that S2(T)5 P2(S). Proof. We consider a factorization
E
S
YE
1.
A c C(K)
'LdI', P )
J,
such that I!Xll llAll 5 P2(S) and y ( K )= 1. Then T := J2AX is a Hilbert-Schmidt operator snd
SAT) 5 P Z ( J 2 ) ll-4Xll
5 P,(S) *
Remark. Using Cadernan's theory of Hilbert-Schmidt operators, cf. [SMI] and [ZAM], we can construct Fredholm divisors for absolutely 2-summing operators.
Distribution of Characteristic Values
27.4.
27.41. We know from 26.5.5 (remark) that every Riesz operator S has a t most a cowtablo set of distinct characteristic values A, with j E J . Put
M := ((i,k): j E J , k
=
1, ...,a(&)] and
:= I$.
Then (Am) with m E M is called the family of characteristic values, counted according t o their algebraic multiplicities. In the following this notation is used without further explanation.
27.4.2. Lemma. Let X E Q(€I,Ii). Then thew is an orthorzormnl furriily (x,~,)i ~ d h nE M such tibut ?bm(S~m, x,) = 1. Proof. Consider the operators A j induced by I - i.,S in N , := N ( ( I - ~,X)"('J)) for j E J . Let {Y(],~): k = 1, ..., .(A,)] be a, basis of 3, such that the representing inatrix (ai,) of A, has superdiagoilal form. Take any order in J , and then use the lexicographical order in M . Construct an orthonormal family (zm)by ap2lying Sclmidt's procedure t o (ym). L c t L,"denote the subspace spanned by (rl,...,xm} or (yl,...,y,]. Since A, is nilpotent. me have = 0 not only for A > I:, but also for 11, k. Hence 1=
k- 1
27. Fredholm Theory
379
This means that (I- 2,s) ym E Lm+.Moreover, we always have (I- 2,s) y, E L,,. Consequently ( I - 2,s) xmE L,-l. Therefore xmis orthogonal to (I-- 2,s) xm. This = 1. proves that Ibm(Sx,,q,&)
27.4.3. Theorein. Let 0 < p
< cw mid 8 E bp(H,H ) . Then
Proof. If 1 s p < ca, then the assertion follows from 15.5.7 snd 27.4.2. For proofs in the remaining case 0 < p < 1 we refer to [GQH, p, 411 and [ZAN, p. $401. Remark. As recently proved by W. B. JORNSON/H. ROXIG/B.MAUREYIJ. R. RETHERFORD [I] the above rzsult can be extended to 9lP-operatorsin Banach spaces. A somewiiat weaker statement, is due to A. S. XARKUS/V.J . NACEAV El]. 27.4.4. As a consequence of 27.3.8 a i d 27.4.3 we obtain the Theorem. Let 0 < r 5 2 and S E %(r,l,2)(E, E ) . Then
37.4.6. Theorem. Let 0
< r 5 1 rrnd S 6 !12(r,l,ll(E, E). Then
5 X{,,,,,,(S), Proof. By 18.1.5 every opwator YP,1,2@)
where
1/p::
I/? -
112.
X E !12(7,1,1)(&,E ) is ( p , 1, ;?)-nuclearand
5 %*I,l)(W*
Remark. Let S,, be the operator in 1: gensrated by the nnitary (n,%)-matrix ,
Then S(r,l,l)(Sn) 5 n*/r-1’2.On the other hand, 8, has n charac-
teristk values with /I,/ = 1. This shows that p is the best possible exponect in the above result.
27.4.6. The next statcirierit follows froiii 27.3.9 am1 27.4.3. Theorem. Let S E ?&(E, E ) . I’hc-n
25.4.7. We now state, without proof, a striking result which has been recently Kijmc/B. MACREY/J.It. RETHERFORD [l]. obtained by W. B. JOHSSOR/€€. Theoreni. Let 2
< r < 00
and S
f
(Pr(E;E). Then
Remark. Since $,(H, H ) =- G2(H,H ) , for 0 cannot be less than 2.
< r 5 2 the exponmt of
con\rergencc
37.4.8. A t this point some results ahout entire functions are collectcd [LEW, p. 211.
Let (A,) with
rtL
M be any family of complex numbers suth that
[Aml-l M
< 00.
380
Part 5. Applications
Then p(A) := R(1 - f./A,)
for all f. E 0
M
is called the canonical prod,&. I n the trivial case 211 = 0 we put p(1) = 1.
Next we observe that the entire function p just defined is of order 1 at most and of mininiuni type. Lemma. Let
> 0. Then there exists e > 0 such that
E
\p(A)l 2 p exp ( E 121) for aEZ iE %?.
Proof. First choose a finitc subset m of M and then g
Since 1
> 0 such that
+ 111. 5 exp ( / A / ) ,we obtain
Theorem. Let d be an entire function with the followingproperties: (1) Given E > 0, there exists e > 0 such that
Id(A)l 2 Q exp ( E 121) for all f. E V. (2) I f (A,) with m E dl denotes the family of zeros, counted according to their order, then 2 lAm1--1 < do. ‘M
(3) d(0) = 1. Under these conditims 42)=
n (1 - ]./Am)
for all 1 E %?.
M
Proof. We know from Hadamard’s factorization theorem [LEW, p. 331 that d ( l ) = ( ~ ( 1 ,exp ) ( a &I where ), p is the canonical product formed with the zeros of d. Since d(0) = p(0) exp ( a ) ,we have oc = 0. The preceding lemma tells us that I, is of order 1 at most and of minimum type. Suppose that #I 0. Then exp (PA) has order 1 and nornial type I@/.By [LEW, p. 211 the product d must be of order 1 and normal type 1/31, as well. But this contradicts (1). Therefore, we have ,8 = 0.
+
+
27.4.9. We next state an important result which improves 27.3.6.
Theorem. Let 8 E !R(,,l,2)(E, E ) . For an arbitrary (1, 1, 2)-nuclear repre+sentation 00
AS’ = 1 u p , ;,
2,
put So := 1 and
1
6, :=
(--l)a 2 ail. . . ui, det ((xfs,aj,>). j,. ....jn=l O0
Then, 00
&A*
d(3.) := 0
for
all Iz E %
27. Fredholm Theory
381
is a Fredblm div-kor of 8.Moreover,
d(1) =
n (1
- A/&,)
for all l E V.
All
Proof. The first part of the assertion follows from 27.3.6. Therefore it remains to show the product representation of d. Without loss of generality we may suppose thah uj 2 0, wz(ai) = 1, and w,(xk) = 1. Then Hadamard's inequality implies
Using 27.4.8 (lemma) we obtain
m
5
n(1-k 0~14)5 e exp
(E
Ill),
j-1
for some p > 0, where E > 0 is given. By 27.4.4, moreover, the conclusion follows from 27.4.8 (t.heorem).
llml-l < 00. Pinally, M
27.4.10. Lemma. (trace (S)l S X(l,l,2)(S)for all S E $(E, E ) . m
Proof. Let S =
uiai @ xi be a finite representation such that 1
According to 18.1.16 we obtain Itrace (#)I
5 N:l.l,dS)
= %,w(S).
By the preceding lemma, the trace admits a unique continuous extension to the E). whole space %(l,l.2)(E, Proposition. Every operator S E %(l,l,2)(E, E ) has a well-defined truce which can be computed by 00
trace (8)= 2' c j ( x j ,aj) 1 W
for
any (1, 1, 2)-nuclear representation S =
dini @
xj.
1
27.4.11. Pinally, we prove a generalization of the Grothendieck-Lidskijtrace formula
Theorem. If S E %(l,l,2)(E, E),then trace (8)= 2 l/lm. M
Proof. We know from 27.4.9 that
382
Pert 5. Applications
and
for any (1, 1, 2)-nuclear represtntat’ion of S. Therefore, the assertion follows iron1 27.4. I0 (proposition). R e m a r k . Since %(l,l,2)(U, H ) = G,(H, U ) ,we obtain RS a special case Lidskij’s tracs formula for nuclear operators acting in a Hilbert space. On the other hand, by %(2,3,1,1) E !Rn,,,,,,,, the result is also true for (2/3, 1, 1)-nuclear operators. In general, the trace formula does not hold for (r, 1, 1)-nuclear operators with 3/3 < r 5 1 ; cf. 10.4.5 (remark).
27.5.
Notes
Determinants of integral operators have been defined by I. FREDHOLX [i]. See also T. CABLE[I]. The abstract determinant theory for nuclear operators in Banach spaces was developed [6], A. F. RUSTON[i], [2], and T. L E ~ A ~ ~Ll]. S KFurther I contributions by A. GROTHENDIECK are due to R. SIKORSKI [i],[2]. In this chapter we have presented a new approach discovered [i].There is an extensive literature dealing with the distributions of charaoby A. PIETSCH teristic values. The first result of this kind was proved by I. SCHUR[I]. Further references are 8. H. CHANG111, K. FAX[l], [2], A. HORN[i],A. PIETSCH [17], and, above all, H. WEYL[I], [2].The so-called trace formula is due to A. GROTRENDIECK [6] and V. B. LIDSKIJ [l]. See also J. A. ERDOS[i].
MAN
Recommendations for further reading:
CGOHI, CRIPU’I, [ S W , CYOSI, [ZANI. C. TV. HA [2], E. HELLINQER/~. TOEPLITZ [i], H. vox KOCH[I], H. KONIG[4], A. S. MARKUS/V.J. MACAEV[l], N. J. XIELSEN111, A. F.RUSTOX[4], [Ti], B. SIMON [i].
25.
Structure Theory of Bnnach Spaces
There is a close relationship between the theory of operator ideals and the geomctry of Banach spaces. In this chapter we only deal with a few aspects.
TTsing sonic results about operator ideals we get deep structure theorem. So the ftmious principle of local reflexivity follows from the fact that B(E,P)' and %(F, E ) are metricdly isomorphic for all finite dimensional Banach spaces E. Moreover, w estimate projection constants by the help of the absolutely 2-summing norm. This yields a short prooI of the fundamental Kadec-Snobar lemma. Opcrator ideals can also be used to characterize classical Banach spaces. Here the quotient of two operators ideals plays a significant role.
Since the coiriponents U ( E ,a) of any normed operator ideal are likewise Banach spaces, we get a, lot of new examples with interesting properties. In particular, we mention the Banach spaces G,(H, K ) ,where 1 p < 00. So, for example, it has b e s observed by Y. GORDONand D. R. LEWISthat %(H, K ) and B(H, 9) do not havelocal unconditional structure.
28.1.
Principle of Local Reflexivity
58.1.1. We start this section with Helly's Lemma. Given M E Dim (El) and E > 0, for every xi E E" there exists ,z, f. E suck thut 1 ) ~ ! 1 2 (1 + E ) IIxJl and (zo,a) = (xi,a ) whenever a E M . Proof. Put
N
:= (z E
E : (z, a) = 0 for a E N )
and
NO0 := (x" E": (x", a) = 0 for a E N}. By B.3.10 we have the diagram
where K is a, metric isomorphism from E / N onto E"INoo. Choose xo E E such4hat K&E,x,,= and [[zo// S (1 e) l[x:II. Then Q$+i - KEzo)= o implies 26 - KEzoE NOO. Hence (ro,a) = (z:, a) for all a E M .
Qs0z{
+
3a
Part 6. Applications
228.1.2. We are nov ready to check a fundamental Lemma. Suppose that E is finite dinzensionaZ. Let S E B(E,F')and M E Dim (P'). Given E > 0, there i s R E E(E,8)such that lIR[l I (1 E ) llS\l and (Rx,b) = (8qb) fordZxEEandbEM. Proof. Observe that
+
M @ E := 1
bi @ x i : xl,..., x, E E,bl, ..., b, E M , n = 1, 2,
...
i q a finite dimensional subspace of % ( F ,E). By 9.2.3 we may identify B(E,P)' and %(F, E ) a8 well as %(F,E)' and B(E, F").Now the assertion follows from Helly's
lemma.
28.1.3. As an improvement of the preceding result we get the principle of local refZezitdy. Lemma. Suppose that E is finite dimensional. Let S E B(E, F")and M E Dim (F'). Given E > 0, then the!reexists R E B(E,F ) such that IlRll 5 (1 E ) 11511,(Rx,b)=(Sx,b) for d x E E and b E M , as well us KFRx = Sx whenever SX E M(KF). Proof. Put Eo := (x E E:SX E M ( K F ) )and denote the injection from Eo info E by J . Choose any surjection Q E B(E, Eo) for which QJ is the identity map of E,. Obviously, x -+ K;'Sx defines an operator So E f?(Eo,F) with llSo(li IlSiI and KpSo = SJ. Let ( M i ) with i E I bc a directed family of subspaces M iE Dim (F')such that F' = U Mi and M i 2 M . By 28.1.2 we can find Ri E B(8,F ) with llRill 5 (1 E ) llSl1
+
+
I
and (Rix, b) = (Sx, b) for all x E E and b E Mi.This yields (RiJx,b) = ( S g ,b) for all x E Eo and b M i . So, according to 9.2.3, the directed faniily (RiJ) tends to So in the weak E(Eo.F)'-topology. Let C be the convex set of all operators RJ, where R E E(E, F ) , such that /]Rll5 (1 E ) ljSjl and (Rx, b} = (Sx,b) whenever x E E and b E M . Clearly RiJ E C. Therefore So belongs to the weak closure of C. Let us recall that we have the same closed convex subsets for the weak topology and the norm topology of B(Eo,P);cf. [DUN, p. 4221. Hence there exists R, E B(E, F ) with RJ E C and ljBoJ - Soil I]&//d E /IS//. Finally, put R := (So- RoJ)Q Roe Then JJRII5 (1 2 ~11 )611. If x E E and b E a,using KFSO = SJ we get
+
+
(AX, bj = (SOQX,bj - (ROJQX,bj
+
+ (Rex, b) = (SX,b}.
Furthermore, Sx E M ( K F )implies z = JQx and therefore
KFRx = KFSOQX+ K,R,(x - JQx)= SX. 28.2.
Geometric Parameters Related to Operator Ideals
28.2.1. Let A be a quasi-norm defined in some operator ideal 31. Then for every finite dimensional Banaoh space E we may consider the value A(IE) which is inva,riant under metric isomorphisms. For certain quasi-norms A it turns out that A(IB) is only a function of dim ( E ) . In other cases A(I,) also depends on the geometric structure of E .
28. Structure Theory of Banach Spaces
385
98.2.2. Wo begin with a trivial example.
Lemma. llI,ll = 1.
28.2.3. As a counterpart of the preceding result we have the Lemma. If dim ( E )= n, then X(IE)= n. Proof. By B.4.8 we can find a finite representation n
ui
IB =
0xi
1
such that IIcci// = 1 and /\sill= 1. Therefore K ( I E )5 n. Conversely, by 6.8.3 and 6.8.5, n = trace (I,) 5 SO(IB) = N(IE).
28.2.4. We now state the most interesting result of t,his section. The following proof was given by S . KWAPIEA. Lemma. If dim ( E ) = 12, then P2(IE)= dZ. Proof. L e t x ~ , . . x,,, . , E E and define the operator X E E(lF, E ) by m
X(&):=
c tixi. 1
Then llXll = wZ(zi). Consider the factorization
I
+ J
where N := N ( X ) and M := M ( X ) ; cf. A.3.4. Clearly H := Zr/N is also a Hilbert %lie. Hence space and dim ( H ) 9 n. By 17.5.3 we have P2(IH)= S 2 ( 1 H )
PAX) 5 IIJx,ll PZ(4d 11&11 5 nl’z IIXII. I t follows from zi = Xei and w2(ei) = 1 that
lZ(xi)= 12(Xei)5 PJX) w2(ei) 5 n%v2(xi). so P2(I,) 5 d ’ 2 . Conversely, using 24.6.5 and 28.2.3 we get n = N(IE)5 P2(IE)2.
* 28.2.5. According to the following characterization &,,(IE) is called the projecEion constant of E. 25 Pic!tsoh. Operator
386
Part 5. Applications
Proposition. Let E be a finite dimensiomlBanuch space. Then L,(Ig) equals the infimum of all positive numbers cr for which the following condition is satisfied: If E, is any Banach space containing E as a subspace, then there existsaprojectimz P E O(Eo, E,) such that E = M ( P ) and III’li 5 u. Proof. For zation
E
> 0 we pnt u := (1 + E ) Lm(IE).By 19.3.9 we can find a factori-
(1x11
IIAJl5 CT. Suppose that E is a subspace of E,. Since L,(O,p) has the with metric extecsioii property, A E O(E,L,(O, p)) extends to A , E O(Eo,L,(Q, p)) such that IiAoii = IIAjl. Then P := J$XAo is a projection on E, for which E = M ( P ) and iJP// u. Hence c satisfies the above condition. Recall that E is a suhspace of Einf:= l,(UEc). If P denotes any projection in Einj with E = M ( P ) ,then we have Lm(IE)2 IlPil. This fact completes the proof.
28.2.6. The following estimate, together with 28.2.5, yields immediately a famous result due to &I. J. K4DEC/&I. G. SNOBAR [I]; cf. B.4.9. Lemma. If dim (E’) = n, then Lm(IE) =( P2(IE)= n1I2. Remark. Let I,, be the identity map of I;. Then, by 9.1.8 and 19.3.10, we have Lm(Ifl)Pl(Ifl)= ‘n. Consequently, it follows from 22.1.1 and 22.1.3 t,hat Jim n-l’zLm(In) = lini dkza,, =~ ~ c ; : . n
So in the above leniina the exponent of n cannot be improved. 28.2.7. The value of the projection constant strongly depends on the geonletry of the Banach space in question. Let us mention the following result; cf. D. J. H. GARLING[IS].
Proposition. Let E be an n-dirtiensitmab Bmach space. Thcn L,(IE) = 1 i/ and only i f E .is metl-ically isomorphic to.:1
28.3.
Isomorphic Charact,crizationsof Classical Banach Spaces
28.3.1. We start this section with a basic result which follows from 19.6.2. Theorem 1. A Banach space 3’ belongs to L, i f and only if @ F ( E ,F ) for all Banach spaces E. By duality we get
&,(E, P)
28. Structure Theory of Banach Spaces
387
Theorem 2 . A Banach space E belmgs to Lp if and only if !&.(E, F ) & S Y ( E , F ) for all Bunach spuces F. 28.3.2. The next criterion is a direct consequencc of 8.2.5 and the previous statement; of. also 17.3.5. Theorem. A Banach s p c e E belongs to L, if and only if F(E, F ) = S ( E ,F ) for all Baimch spaces F. 28.3.3. We now formulate, without proof, a similar result which was obtained by
D. R. LEWIS/C.P. STEGALL[l]. For this purpose let us recall that F, is the ideal of all Banach spaces having the lifting property; cf. C.3.8. if and o d y if F ( E , F ) = R ( E , F ) for
Theorem. A Banuch spuce E belongs to F;*' all B a d spaces P .
28.3.4. In the following H denotes the ideal of all Banach spaces isomorphic to some Hilbert space. Remark. Obviously, by 19.3.8, we have H = L,. 28.3.5. Next we state a special case of 28.3.1. Theorem. 1. A B a m h space F belongs to for all Banach spaces E .
H if and only if !#P1(E, P ) V2(E,F )
The dual version looks as follows.
F) Theorem 2. A Banach space E belongs to H if and only if !&(E, F ) E Vgua1(E, for aU Banach spaces F . 28.3.6. The following open problem is closely related to some results of Section 17.6. Conjecture 1. A Banach space F belongs to H if and only if @ p l ( E , P )E a 2 ( E , P ) for all Banach spaces E. Conjecture 2. A Banach spacl: E belongs to H if and only if V P , ( EF) ,
s S z ( E ,F )
for all Bansch spaces F.
28.3.7. The next question arises from 11.5.2 and 11.6.2. Conjecture 1. ABanach spaceFbelongs to H ifandonlyif S2(E,F ) = U","(E, F ) for all Banach spaces E . Conjecture 2. A Banach space E belongs to H if andonlyifU,(E, P)= U;"j(E,F ) for all Banach spaces F .
28.3.8. We now prove another basic result. Theorem. f i t I 5 p
2 5q5
00.
.F
Then H = L F n L y .
Proof. Since the normed operator ideal $ is maximal, it follows from 22.1.5 that H & L F . By duality we get H Ly.This proves that H LinJ n L r . 25*
Port 5 . Applications
388
Lp Ly.
Conversely, suppose that E’ n Then JE E &, and QE E f&. Hence 22.5.1 yields JzISl&BE 4. Finally, using the injectivity and the surjectivity of $jwe obtain I E E .Sj and therefore E E H.
28.3.9. As an immediate consequence of 21.4.6 we get a mpplement of the preceding result. Theorem. Let 0 < p
< 00. Then H == C(2,p)n T(2,ph.
28.3.10. Theorem. The Banach space F belongs to H if ultd onlg i f 2(Z17 F ) = g(ll,F ) . Proof. By 22.4.4 we have 2(Z1, F ) = ?J3(Zl, F ) whenever F E H. Conversely, let us suppose that all operators S E .(?(I,, F ) are absolutely sumnung. Then 2(Zl,F ) = p2(Z1, F ) . It turns out that we also have .(?(Zl(I), F ) = ?J32(Z1(I)7 F) for every index set I . Since Fur :=Z1(UF),we get Q p E clg,(FsUr,P ) B(Fs”‘, F ) . Now the surjectivity of 8 implies IT E .Sj and therefore F E H. Remark. As recently shown by S. V. KISLJAKOV [1] and G. PTSIER [4]it does not follow from B(E,E,) = ?J3(E,12) that E belongs to L,.
28.4.
The Structure of Certain Ideal Components
28.4.1. Let [U, A] be a normed operator ideal. Then U(E, F ) is also a Banach space, and we may ask “HOWcertain properties of E and F are transmitted to U(E, F)?” In particular, it is interesting to know for which Banach spaces E and F the corresponding coniponent U ( E ,P ) belongs to a given space ideal B. The following necessary condition can he easily checked. Proposition. #uppose that E + (0) and F =/= ( o f . Them %(E,F ) E B implies E E Bdual and 3’E B. Proof. Take yo E P and 6, E F‘ such that yo + o and (yo, 6,) = 1. By setting Ja := a tgyo for a E E‘ and &[S]:= S’b, for S E U(E, F ) we obtain operators J O(E’, U(E,F ) ) and Q E O(%Z(E,P),E ) with IE, = &J. Therefore E’ U(E,F ) by 13.4.4.So U(E,P ) E B implies E‘ E B. The conclusion F E B can be proved with the same method.
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[3] Finite representability a.nd super-ideals of operators, Dissertationes Xath. f4]Closed operator ideals and interpolation, J. Functional Analysis. HGIXRICH, SJPIETSCH, A. [l] 4 , characterization of (ce,p, q)-integral operators, Math. Nachr.
ISMAGILOV, R. 8. [3] Diameters of compact sets in linear normed spaces (Russian), Collection of papers on “Geometry of linettr spaces and operator theory”, pp. 75-113, Jaroslavl 1977. JOHNSON, W. B./KoxiG, H./XAUKEY,B./RETHERFORD, J. E. [l] Eigenvalues of p-suiuming and $-type operators in Banach spaces, J. Functiorml Analysis. KISLJAKOV, S. V. [2] p-absolutely summing operators (Russian), Collection of papers on ‘‘Geometry of linear spaces and operator theory”, pp. 114- 174, Jaroslavl 1977. liOXIG, IT. [ 5 ] :Interpolation of operator ideals with an application to eigenvalue distribution problems, Mat,h. Ann. 233 (1978), 35-43. [6] Approximat.ion numbers of Sobolev imbeddings over unbounded domains, J. Functional Analysis. [7] s-numbers of Beaov-Lorentz imbeddings, Math. Nachr. [8]‘4formula for the eigenvalues of a compact Operator, Studia Math. [9] s-numbers, eigenvalues, and the trace theorem in Banach spaces, Studia Xath. 1;lO’j On the spectrum of products of operator ideals, Math. Nachr. K W A P I E ~S./PELCZYRSKI, , A. [2] Remarks on absolutely summing translation invariant operators from the disc algebra and its dual into a Hilbert space, Michigan Xath. J. KURSTEN,K.-D. (21 About some questions of A. Pietsch (Russian), Teor. Fonkcii Funkcional. Anal. i Prilofen. 28 (1957), 45-51; 29 (1978), 61-73; MR 66 # 3665. LEWIS,D. R. [5] Finite dimensional subspaces of Lp, St,ndia Xath. 63 (1978), 207-212. [6] Ellipsoids defined by Banach ideal norms, Xathematika. L I S U E , I\?. [fi] Operator ideals and vector measqres, Proc. of the Conf. “Measure and topoygol 1975. [6] Xappings of vector valued measures, Ann. Univ. Craiova. .>n
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Bibliography
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PIETSCH, A. [22] Recent progress and some open problems in the theory of operator ideals, Proc. Intern. Conf. “Operator algebras, ideals, ...”, Teubner-Texte Math., pp. 23-32, Leipzig 1978. [23] Rosenthal’s inequality and its application in the theory of operator ideals, Proc. Intern. Conf. “Operator algebras, ideals, ...”, Teubner-Texte Math., pp. 80-88, Leipzig 1978. [24] Products and quotients of closed operator ideals, Proc. Third International Symposium in Yugoslavia on topology and its applications, Beograd 1977. [25] Distribuhion of eigenvalues and nucle,arity, Banach Centre Publications VIII. [26] Factorization theorems for some scales of operator ideals, Math. Nachr. J. Functional Analysis. [27] Inessential operators in Banach spaces, Integral Equa. and Op. Theory, 1 (1958), 589-591. PISIEX,G. [4] Une nouvelle classe d’espaces de Banach verifiant le thhor6me de Grothendieck, Annales de 1’Inst. Fourier, 28 (197S), 69-90. [ 5 ] Grothendieck’s theorem for non-commutative C*-algebras with an appendix on Crothendieck’s constants, J. Functional Analysis, 29 (1978), 397-415. [6] Some results on Banach spaces without local unconditional structure, Compositio Math. 37 (1978), 3-19. [7] Un nouveau thb&me de factorisation, C. R. Acad. Sci. Paris, SC. A, 286 (1977), 715-718. Pum, J. [2] Selfadjoint and completely symmetric operator ideals, Math. Nachr.
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Index of Axioms
(01,)
operator ideals
(XI,) sequence ideals (FI,) (SI,) (QOI,) (QNI,) (QFI,)
family ideals space ideals quasi-norms of operators quasi-norms of sequences quasi-norms of families (OS,) s-numbers of operators (NS,) s-numbers of sequences
1.1.1, 29.2.1 13.1.1 16.1.1 2.1.1, 29.2.2 6.1.1 13.2.1 16.1.3 11.1.1 13.7.2
Index of Operator Ideals
Operator ideals and their quasi-norms are denoted by capital bold Gothic and capital bold Roman letters, respectively. The corresponding space ideal is marked by capital bold sanserif letters. Example : operator ideal a, quasi-norm A, space ideal A.
Symbol
Ideal of
...
Gp)-operators operators of ( 8 , p)-cotype dualisable operators p-dominated operators ( p , q)-dominated operators Gp-operators finite operators * discretely p-factorable operators 29*
Point 14.2.4 21.4.1 4.4.10 17.4.7 17.4.1 14.3.1
B.hl 19.3.11
440
Symbol
Index of Operator Ideals
Ideal of
...
approximable operators @(*)-operators Hilbert operators N,-Hilbert operators Gr)-operators integral operators r-integral operators ( r ,p, p)-intogral operators o-integral operators compact operators p-compact operators ( p , q)-compact operators arbitrary operators p-factorable operators ( p ,p)-factorable operators ( 8 , p)-mixing operators nuclear operators r-nuclear operators ( r , p , q)-nuclear operators o-nuclear operators strictly nuclear operators absolutely summing operators absolutely r-srxniming operators absolutely ( r , p)-suu”irningoperators absolutely ( r ,p , p)-summing operators absolutely z-summing operat>ors decomposing operators p-decomposing operators inessential operators p-radonifying operators strictly singular operators Gp-operatorson Hilbert spaces GF)-operators Gf)-operators strictly cosingnlar operator operators of (8, p)-type unconditionally suinming operators operators of ultra (8,p)-type conipletely continuous operators weakly compact operators separable operators Radon-Nikodym operators
Point
1.3.1 14.1.11 6.6.1 5.4.1 14.2.15 6.4.1 19.2.1 19.1.1 23.3.1 1.4.1 18.3.1 18.4.1 A.3.2 19.3.1 19.4.1 20.1.1 8.3.1 18.2.1 18.1.1 23.2.1 18.7.1 6.5.1 17.3.1 17.2.1 17.1.1 23.1.1 24.42 24.5.3 4.3.5
25.4.2 1.9.2 15.5.1 14.2.1 14.1.1 1.10.2 21.2.1 1.7.1 21.3.1 1.6.1 1.5.1 1.8.1 24.2.6
Index of Family Ideals
Family ideals and bheir qtiasi-norms are denoted by small bold Gothic and small bold Roman letters, respectively. Example : family ideal a, quasi-norm a.
...
Symbol
Ideal of
I
bounded families absolutely p-summable families mixed (s,p)-summable families weakly p-summable families
1, m(8jP)
5
Point A.4.1 16.2.1 16.4.1 16.3.1
Index of Scalar Sequence Ideals
Scalar sequence ideals and t,heir quasi-norms are denoted by small Gothic and small Roman letters, respectively. Example : scalar sequence ideal a, quasi-norm a.
Symbol CO
i 1 1, I,,.,, I@ Gai
Ideal of ...
Point
zero sequences finite sequences bounded sequences absolutely p-summable sequences ( p ,q)-Lorentz sequences Q-Orlioz seque'nces @-Sargent sequences
13.1.6 13.1.4 A.4.1 13.2.10 13.9.1 13.11.1 13.10.1
Index of Procedures
~~
Symbol
Procedure
Point
clos rad dual reg inj sur max min
closure radical dual ideal regular hull injective hull surjective hull maximal hull minimal kernel adj oint ideal
4.2.1 4.3.1 4.4.1, 8.2.1 4.5.1, 8.3.1 4.6.1, 8.4.1 4.7.1, 8.5.1 4.9.2, 8.7.1, 13.5.1 4.8.2, 8.6.1, 13.4.1 9.1.1, 13.6.1
*
Index of Extensions
Symbol
Extension
Point
SUP inf mP rin lUP lin
superior inferior right-superior right -inferior left-superior left-inferior
15.6.2 15.6.4 15.6.6, 29.5.4 15.6.8 15.6.10 15.6.12
Index of Symbols
Fixed ndatim coniplex field n-dimensional complex linear spaoe Banach function space; C.1.5 Banach function space; 25.5.1 Banach sequence space; C.1.7 set of all e = (el, .,E,) with ~i = f1 ; E.5.1 single unit family (1) i-th unit family characteristic family of M Kronecker's symbol normalized n-dimensional Gauss probability chracteristic function of B injection defined by n ;A.4.3 real or complex field n-dimensional real or complex linear space Banach function space; C.1.2 Banach sequence space; (2.1.3 Banach vector space; C.1.3 stable law; 21.1.1 %-dimensionalstable law; 21.1.3 set of natural numbers [ 1 , 2 , . .) n-dimensional canonical projection surjection defined by n ; A.4.3 real field set of non-negative real numbers n-dimensional real linear space normalized rotation invariant measure on the n-dimensional sphere closed unit ball of lp closed unit ball of 1; Banach function space; 22.7.1
..
.
Let E be any Banach space LT.3
R' E" p n j
Fur 1,
closed unit ball; A.1.3 dual Banach space ; A.2.3 bidual Banach space; 4.2.4 Banach space I,( UEp) ; (3.3.3 Banach space ll(UE);C.3.7 identity map; A.1.4
444
Index of Symbols
KE JE QE
dim ( E ) Dim ( E ) Cod ( E ) Car (3)
W)
evaluation map ; A.2.4 canonical injection froin ik: into EinJ; C.3.3 canonicnl surjection from E'"' onto E ; C.3.7 dimension of E collection of all finite diiaensiond subspaces collection of all finite codiinensional subspaces collestion of Cartesian Banach spaces; 5.3.7 algebra of cylindrical subsets
Let LV aid N be subspccs of E dim (N) cod
(w)
J% 0%
8N(E)
dimension of M oodimension of N embedding map froin ill into E ; A.1.5 canonical map froin E onto EIN; A.1.6 o-algebra of cylindrical subsets
Let E and F be Baitack spaces
2(E,F )
EXF E
natural pairing; A.2.1 1
(a,
445
446
Index of Symbols
Universal constants anp
CG CK CP CSP
p) 89
quotient c!$/cg); 22.1.1 Grothendieck’s constant,; 22.4.5 converse Khintchine’s constant ; E.5.3 Khintchine’s constant; E.5.2 absolute p-th moment of pus;21.1.2 absolute p-th moment of p ; ; 22.1.1
Index of Subjects
absolutely - p-summable family 16.2.1 summing operator 6.5.1 - r-summing operator 17.3.1 - (r,p)-slimming operator 17.2.1 - (7, p, q)-summing operator 17.1.1 - t-summing operator 23.1.1 abstract - Lp-space C.2.1 Loo-space C.2.3 additive s-function 11.8.1 adjoint - operator D.2.2 - operator ideal 9.1.1 - sequence ideal 13.6.1 a,lgebraicmultiplicity 27.1.3 approximable operator 1.3.1 approximation number - of an operator 11.2.1 - of a sequence 13.7.3 approximation property 10.1.1 ascent 26.1.2
-
-
Banach - space A.1.2 - space idea,] 2.1.1 basis 23.2.3, 28.5.1 bidual - Banach space A.2.4 - operator A.3.6 Bochner integral 24.1.9 Bore1 - probability C.1.8, 25.1.2 - a-algebra (2.1.8, 25.1.1 Brownian mot,ion 25.6.4 Calkin algebra 26.3.4 Cartesian - Banachspace 2.1.4 - operator ideal 29.4.1 - space ideal 29.4.2 Cartesian product - of Banach spaces B.4.11 - of operators 29.1.11 ’ - of locally convex spaces 29.1.10
Ctlrtesian $,-product C.4.1 characteristic - family A.4.4 - value 26.5.4 closed operator ideal 4.2.4 closure of an operator ideal 4.2 1 compact operator 1.4.1 p-compact operator 18.3.1 (p, 9)-compact operator 18.4.1 complemented subspace B.4.3 completely continuous operator 1.6.1 completely symmetric - operator ideal 4.4.5 quasi-normed operator ideal 5.2.4 - 8-function 11.7.1 COtYpe Banach space of (s,p)-cotype 21.4.11 operator of (sr,p)-cotype 21.3.1 cylindrical - probability 25.3.1 - Gauss probability 26.5.3 - Radon probability 25.3.5 - set 25.2.2
-
decomposable operator 24.2.1, 24.4.1, 245.1 decomposing operator 24.4.2 p-decomposing operator 24.5.3 desuent 26.1.3 dimension of a Hilbert space D.1.3 direct sum B.4.2 discretely p-factorable operator 19.3.11 dominated operator p-dominated operator 17.4.7 (p, q)-dominated operator 17.4.1 domination property - of a norrned operator ideal 15.4.1 - of a normed sequence ideal 13.8.1 dual - Banach space A.2.2 - exponent C.1.4 - operator A.3.5 - operator ideal 4.4.1 - quasi-normed operator ideal 8.2.1 - space ideal 4.44 dualisable operator 4.4.10
448
Index of Subjects
Dunford-Pettis property 3.2.8 -- operator 3.2.5
-
elementary operator B.1.5 entropy number -, inner 12.1.6 -, outer 12.1.2 equivalent quasi-norms 6.1.7 evaluation map A.2.4 extension property C.3.1 extension of an operator ideal from Banach spaces to locally convex spaces 29.5.1 - from Hilbert spaces to Banach spares 15.6.1 factorable operator A-factorable operator 2.2.1, 29.2.5 p-factorable operator 19.3.1 (p, q)-factorable operator 19.4.1 family A.4.1 family ideal 16.1.1 finite ascent 26.1.2 - descent 26.1.3 operator B.l.l - sequence 13.1.4 finite dimensional Banach space 2.3.1 finite nuclear norm 6.8.1 Fredholm coefficient 27.2.4 determinant 27.2.7 -- divisor 27.1.5 minor 27.2.8 - operator 26.3.1 - resolvent 26.2.2 fractional integration 22.7.5 function s-function of operators 11.1.1 s-function of sequences 13.7.2 functional A.2.1
-
-
-
Gauss probability -, cylindrical 25.5.3 -, normalized cvlindrica! 25.5.2 Gelfand number 11.5.1 Gohberg operator 4.3.5 Grothendieck - constant 22.4.5 - operator 3.2.6 space ideal 29.6.1
-
Hadamard inequality 27.2.2 Hilbert -- number 11.4.1 - operator 6.6.1 - space D.l.l N,-Hilbert operator 5.4.1 Hilbert-Schmidt operator 15.5.5 HBlder - space 22.7.1 - A-continuous function 26.6.1 hull procedure - on the class of operator ideals 4.1.2 - on the class of quasi-iiormed operator ideals 8.1.2 - on the set of quasi-nornied sequenre ideals 13.3.2 hyperorthogonal basis 23.2.3 idem potent - procedure 4.1.2 - operator ideal 3.1.5 -- quasi-normed operator ideal 7.1.5 identity map A.1.4 inemential operator 4.3.5 inferior extension 15.6.4 injection B.3.2 @-injection 26.6.1 injection modulus B.3.1 injective -- operator ideal 4.6.8, 29.3.1 - quasi-normed operator ideal 8.4.8 - space ideal 4.6.8,29.3.2 - s-function 11.5.4 injective hull - of an operator ideal 4.6.1 - of a quasi-normed operator ided 8.4.1 inner entropy number 12.1.6 integral operator 6.4.1 r-integral operator 19.2.1 (P, p, q)-integral operator 19.1.1 a-integral operator 23.3.1 intermediate space C.5.3 isomorphic - Banach spaces B.2.2 - locally convex spaces 29.1.8 isomorphism - between Banach spaces B.2.1 - between locally convex spaces 29.1.8 @-isomorphism 26.3.1 Eat0 operator 1.9.2 kernel 24.2.1, 24.4.1, 24.5.1
Index of Subjects ~
449
~~
kernel procedure - on the class of operJtor ideals 4.1.3 - on the class of quasi-normed operator ideals 8.1.2 - on the set of quasi-normed sequence ideals 13.3.2 Khintchine - constant E.5.2, E.5.3 - inequality E.5.2 Kolmogorov number 11.6.1 left-inferior extension 15.6.12 left-superior extension 15.6.10 lift.ing property C.3.5 limit order 14.4.1 locally convex space 29.1.1 locally convex space ided 29.2.2 local unconditional structure 23.3.7 Lorentz sequence ideal 13.9.1 maximal - operator ideal 4.9.6 - quasi-normed operator ideal 8.7.7 - quasi-normed sequence ideal 13.5.5 - 8-function 11.10.1 maximal hull - of an operator ideal 4.0.2 - of a quasi-normed operator ideal 8.7.1 - of e quasi-normed sequence ideal 13.3.1 metric - approximation property 10.2.1 - extension property C.3.1 - injection B.3.2 - isomorphism B.2.1 - lifting property (3.3.5 - surjection B.3.6 metrically isomorphic Banach spnce €3.2.2 minimal - operator ideal 4.8.6 - quasi-normed operator ideal 8.6.5 - quasi-normed sequence ideal 13.4.5 minimal kernel - of an operator ideal 4.8.2 - of a quasi-normed operator ideal 8.6.1 -- of a quasi-normed sequence idonl 13.4.1 mixed (8, p)-sumniabk fdnnlj 16.4.1 mixing operator (8, p)-mixing operator 20.1.1 moment -, absolute p-th 25.3.10 -, week p-th 23.3.8
monotone procedure 4.1.2 multiplicative 8-function 11.9.1 Noether operator 26.3.1 non-increasing rearrangement 13.7.6 norm - of family ideal 16.1.7 - of an operator ideal 6.2.1 - of a sequence ideal 13.2.6 normed -- family ideel 1 6 . 1 3 - operator ideal 6.2.2 - sequence ideal 13.2.7 nuclear locally convex space 29.7.1 nuclear operator 6.3.1 r-nuclear operator 18.2.1 (r,p, p)-nuclear operator 18.1.1 u-nuclear operator 25.2.1 null space 8.3.4 niim ber 8-number of operators 11.1.1 8-number of sequences 13.7.2 operator A.3.1, 29.1.6 Ep-operator 14.2.1 &p-operator 14.3.1 gp-operntor 14.2.15 Gp-operator 15.5.1 G;)-operator 142.1 @)-operator 14.1.1 operator ideal - on Banach spaces 1.1. L - on locally convex spew 29.2.1 Orlicz property 17.2.8 Orlicz sequence ideal 13.11.1 orthogonal projection B.4.6 outer entropy number 12.1.2 partially isometric operator D.2.4 Pelczyriski - property 2.0.1 - operator 1.10.2 perfect - normed operator ideal 9.3.3 - normed sequence ideal 13.6.6 positive operator D.2.3 procedure - on the class of operator ideals 4.1.1 - on the class of quasi-normed operator iC!eals 8.1.1 *. - on the set of q:iasi-normcd sequence ideals 13.3.1
450
Index of Subjects
product - of operator ideals 3.1.1 - of quasi-norms 7.1.1 projcction B.4.1 projection constant 28.2.5 proper - operator ideal 2.3.3 - sequence ideal 13.1.7 pseudo-s-function 12.1.1
right-inferior extension 15.6.8 right-superior extension 15.6.6 Rosenthal operator 3.2.4
Sargent sequence ideal 13.10.1 Schmidt factorizatiori D.3.3 - representation D.3.2 Schur property 2.5.1 quasi-compact operator 26.4.1 Schwartz space 29.6.10 quasi-dual Banach space 4.4.9 self-adjoint quasi-norm - norrned operator ideal 9.4.1 - of a family ideal 16.1.3 - normed sequence ideal 13.6.8 - of an operatm ideal 6.1.1 separable -- of sequence ideal 13.2.1 - Banach space 2.7.1 quasi-normed - operator 1.S.l - family ideal 16.1.5 sequence ideal 13.1.1 - operator ideel 6.1.3 simple Banach space 5.2.1 - sequence ideal 13.2.3 small operator ideal 14.5.1 quotient space A.1.6, 29.1.5 space quotient L -s ace 22.2.1 p. - of operator ideals 3.2.1 space ideal 2.1.1, 29.2.2 - of quasi-norms 7.2.1 Bobolev Rademacher type limit order 22.7.3 Banach spaces of Rademacher (8, p)-twypvpe - space 22.7.1 21.2.12 Sobolev-Slobodetzkij space 22.7.1 operators of Rademacher (8, a)-type stable law 21.1.1 21.2.6 strictly cosingular operator 1.10.2 radical 4.3.1 strictly nuclear Radon probability 28.1.2 - locally convex space 29.8.1 radonifying operator - operator 18.7.1 p-radonifying operator 25.4.2 strictly singular operator 1.9.2 Radon-Nikodym strongly - derivative 24.3.4 - bidual operator 1.5.5 - operator 24.2.6 - p-factorable operator 19.3.11 -- property 24.2.9 - r-integral operator 19.2.15 range A.3.4 subspace A.1.5, 29.1.4 rank B.l.l summable family A.4.6 reflexive Banach space 2.4.1 superior extension 15.6.2 regular surjection B.3.6 - Bore1 probability C.1.8, 25.1.2 @-surjection 26.6.6 - operator ideal 4.5.5 surjection modulus B.3.4 - quasi-normed operator ideal 8.3.4 surjective - s-function 11.7.1 - operator ideal 4.7.8, 29.3.5 regular hull - quasi-normed opeartor ideal 8.5.6 - of an operator ideal 4.5.1 surjective - of a quasi-normed operator ideal 8.3.1 - space ideal 4.7.8, 29.3.6 related operators 27.3.1 - s-function 11.6.4 Riesz surjective hull - decomposition 26.1.5 - of an operator ideal 4.7.1 - operator 26.5.1 - of a quasi-normed operator ideal 8.5.1
-
-
Index of Subjects symmetric operator ideal 4.4.5 - s-function 11.7.1
-
trace B.1.4 trace formula 27.4.11 type Banach space of (s,p)-type 21.2.12 operator of (8,p)-type 21.2.1 operator of ultra (s,p)-type 21.3.1 type of an intermediate space J-type C.5.6 K-type C.5.4 ultraproduct - of Banach spaces 8.8.1 - of operators 8.8.3 ultrastable - quasi-normed operator ideal 8.8.5 - 8-function 11.10.5 unconditionally summing operator 1.7.1 unit family A.4.4
451
viiriation 24.3.2 vector function 24.1.1 -, p-integrable 24.1.9 -, p-measurable 24.1.3 -, ,u-simple 24.1.2 vector measure 24.3.1 -, p-continuous 24.3.4 -, p-differentiable 24.3.4 weak operator topology A.3.3 weak topology A.2.3, A.2.5 weakly compact operator 1.5.1 weakly lower semi-continuous quasi-norm of an operator ideal 8.7.16 quasi-norm of a sequence ideal 13.5.7 - s-function 11.10.7 weakly singular integral operator 22.7.5 weakly p-sunimable family 16.3.1 Wiener probability, cylindrical 25.6.4
-
