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0 let us select the numbers {e
k-l
1+e
I
in (5.2) by every block set {y }
i=l
> 0 } _ , with
k-l
£ n (l+e ) 11
c S(B) of
i
and
1-e
^
n (1~© )• " i=l
i
Let
L c 0
B be
an
N -dimensional k
Milman & Perelson: Geometric moduli
subspace with N
35
large enough (we may compute t h i s
k
N
function of N(m;N
as a
k-iterated
k
2
; 1+ - ;e ) where N(m,n,C,e) is defined in Proposition
5.2). Let x € S(L ). Using Proposition 5.2, we select a subspace L c L , dim L = N
, such that for every y € L
(5.3)
(l-e1)(l+6m(|y||;xi))^ fx^yl * (1+e^ ( 1 + M ||y||; x ^ ).
We continue by induction. Suppose that the elements {x > were L
already
c L
n-l
constructed
with
. As in the first
n-2
subspace L c L n
, dim L
n-l
the appropriate
step,
= N
n
for every y € L and x € S(span
n
of subspaces
) one can choose a
n-1
= N(m;N ; 1+- ;e ), such
k-n+1
k-n
e
n
n
that
{x > n ) we have
n
(5.4)
for x € S(L
and N
k-n
sequence
j 1
(l-en)(l+Sm(||y|;x))
s |x+y| * (1+oJ ( 1 + Mflyfl;x ) ) .
Repeating t h e use of ( 5 . 4 ) we o b t a i n , f o r a =1,
c-1
I
)
x aa. x J
Lj=1
J J
,,rk-l
"
k
I^I
E
ax
ax
r
II) •_ \
L Lj=l
'
^ II
ax ;
a
I
; k>—a1 x
\
^m ~k~T
^
||
iirk-l
I
C
+
II r
j=l
+ |,
II) _ k —1
(1+G
ax
J
jXjH
* ... ^
a y
E
-
j=l
J M
—
)
^ (l+e) n L
This proves t h e r i g h t left
; [ax
1+ P
j=2 I
m
l y3"1 a x I
L
Ln=l
n n"
hand s i d e of ( 5 . 2 ) .
L nn n=l
.
lI J J
In a s i m i l a r
way we p r o v e t h e
hand s i d e of ( 5 . 2 ) . Note
that
by c o n s t r u c t i o n
we may r e p l a c e
{x}
i n ( 5 . 2 ) by
36
Milman & Perelson: Geometric moduli 1*1
every
block
sequence
iy« =
) J=n
a x
>
, o f { x }
(since
the
sequence
of
£+1 n
subspaces {L.}
is a decreasing sequence and span {y«}^_ c span {x } s+ ).
We return to the proof of Theorem 5.1. Let T),K > 0 be given. There exists a positive integer m such that
(5.5) and \fi $ (7)) ~ BBit))\ 'mm 00
'
< K.
The functions (B (A;x) and 8 (A;x) are Lipshitz on [0,1] x S(B). Then, by [Ml] (Theorem 5.1), for any N, there exists an N-dimensional subspace L c B such that for every A€ [0,1] the oscillation of the two fuctions £ U;x) m
and
5 (A;x) on S(L ) is small (say, less J m o Proposition 5.3 with this subspace L , then
than
a
&given
e>0). Use
j-i (5.6)
1+5 h ; m L
V a x I n n J n=l
£ (l-e)(l+
L
sup 5 ( T ) ; X ) ) fc ( 1 - e ) ( 1 + 5 5 ( T J ) ) . m m x€S(L ) m 0
Suppose that for a given k we select € in such a way that
(l-e) k ^ 1/2. Using similar arguments to those in section 3 together with the left hand side of (5.2) and (5.6) we obtain
t
5.7)
k+1 rp i V n (1+6 5 (77)) < -=- 3T (1 + (3T T))p) .|J mm [l-Q pj p
for some integer function t such that t k
p
> oo (if k—^co). k
Milman & Perelson: Geometric moduli
37
Assume that K has been chosen in such a way that (1-K) k £ 1/2.
Using (5.5) and (5.7) we have r >,
t
(1+5 5 (7))) oo
^
t /p
^
T~T: ^^
(1 + C3T T)) ) . pj p
^1-e
Taking the t -th root from both sides of the above inequality we obtain (for t=t ) k
1/t
l/p
(1 + (3TT>) P ) P
*
l/t
* j^l-e Ui- 3Tpj Let t—>oo,
(5.8)
(1 + (3T T))p).
then
5 5 (TJ) ^ (3T TJ 0 0
p
and 5 6 (3T ) p .
^ p
The p r o o f
p
of
the
formula
for
(3(3 ( c )
is
similar.
We
obtain
f 1 1q ( 5
'
9 )
3^
and then take the lim inf. •
As a corollary of Theorem 5.1
(or, to be more exact, from
(5.8) and (5.9)), we obtain Theorem 2.3 stated in section 2 in the same way as we proved Theorem 2.1 at the end of section 3.
38
Milman & Perelson: Geometric moduli REFERENCES.
[Cl]
Clarkson, J.A. (1962). Orthogonality in normed linear spaces. Archiv Math. 4:4, 297-318. Day, M. M. (1941). Reflexive Banach spaces not isomorphic to [D] uniformly convex spaces. Bull. Amer. Math. Soc. 47:4, 313-317. [K] Kadee, M.I. (1956). Unconditionally convergent series in a uniformly convex space. Uspekhi Mat. Nauk 11:5, 185-190 (Russian) [Li] Lindenstrauss J. (1963). On the modulus of smoothness and divergent series in Banach spaces. Michigan Math. J. 10:3, 241-252. [Ml] Milman, V. (1971). Geometric theory of Banach spaces II. Geometry of the unit sphere. Uspechi Mat. Nauk, 26 no. 6, 73-149; Russian Math. Surveys 26, 6, 80-159. [M2] Milman, V. D. (To appear),The heritage of P. Levy in geometric functional analysis. Asterisque. [M3] Milman, V. D. (1967). Infinite-dimensional geometry of the unit sphere of a Banach space. Soviet Math. Dokl. 8, 1440-1444 (Translated from Russian). [MaPi] Maurey, B. & Pisier, G. (1976). Series des variables aleatoires vectorielles independantes et priorietes geometriques des espaces de Banach. Studia Math. 58, 45-90. [Mil] Milman, D.P. (1938). Certain tests for the regularity of spaces of type B. Dokl. Akad. Nauk SSSR 20, 243-246 (Russian). [MSch] Milman, V.D. & Schechtman, G. (1986). Asymptotic theory of finite dimensional normed spaces. Springer-Verlag. Lecture Notes in Mathematics 1200, 156 pp. [MShl] Milman, V.D. & Sharir, M. (1979). A new proof of the Maurey-Pisier theorem. Israel J. of Math. 33, 1, 73-87. [MSh2] Milman, V. D. & Sharir, M. (1979). Shrinking minimal systems and complementation of n -spaces in reflexive Banach spaces. p
Proceeding of London Math. Soc., (3) 39, 1-29. [Sm] Shmul'yan, V. L. (1940). On the differentiability of the norm of a Banach space. Dokl. Acad. Nauk SSSR 27, 643-648.
HUBERT SPACE REVISITED.
J. R. Retherford Louisiana State University Baton Rouge, LA 70803 Dedicated to Professor Antonio Plans
1. INTRODUCTION, Since some of the work of A. Plans concerns operators on Hilbert
spaces
(e.g.
[PI
1-5], [PB])
it
seems
appropriate,
in
this
"Festschrift" on the occasion of the retirement of Professor Plans, to look back at a few (of many) operator characterizations of Hilbert space. We will show that all of these characterizations follow easily from the summability property of the eigenvalues of nuclear operators on isomorphs of Hilbert space. This property is discussed below.
2. DEFINITIONS. Recall that an operator T from a Banach space X to a Banach space Y, here after written T : X —> Y, is nuclear provided there are (fn ) * in X , (y ) in Y such that
Tx = ) f (x) y L n n n=l
for each x € X, and
I « f n i \WJ < -• This concept is due to Grothendieck [G3] and Ruston [Ru]. On Hilbert
spaces the nuclear operators
"trace-class" operators if , where T e if means
coincide
with the
Retherford: Hilbert space
40
n=l Here (A v TT ) denotes the eigenvalues of the unique positive square root * n of TT . Moreover, it is readily seen that
1
2
2
where !f denotes the Hilbert-Schmidt class. That is, each nuclear operator on a Hilbert space is the composition of two Hi lbert-Schmidt maps and, conversely, the composition of two such maps is nuclear. The
generalization
of
this
fact
to
Banach
spaces
(with
applications) is the subject of this paper. The Hilbert-Schmidt operators have been generalized to Banach spaces
in the following
way
[Gl], [Pi3],
[S]: An operator
T: X—>Y
is
absolutely two summing, written T € TT (X,Y), provided there is a constant K such that
for all finite sets x ,...,x 1
in X. n
Pietsch [Pi3] has shown that for Hilbert spaces H , H
so TT is a natural generalization of the Hilbert-Schdmidt class. The class TT , defined by replacing 2 by p, 1
p < +oo, above,
p
has seen numerous applications in Banach space theory. The
famous
Grothendieck-Pietsch
factorization
theorem
[Gl]
[Pi2,5] asserts that T€ TT (X,Y) if and only if T admits the factorization
41
Retherford: Hilbert space
where K is a compact Hausdorff space, ji a normalized Borel measure on K, and j the inclusion mapping. It is easily seen that j € II (C(K), L (JI)). We will also have occasion to use the p-nuclear operators, in what follows: T is p-nuclear, 1 ^ p < +00, written T € N (X,Y) if p
Tx = V f (x)y L
n
n
(f ) in X , (y ) in Y, x in X where
and
< +00
" «
In
particular
V |f(y ) |q
Sup
llflhl
L
< +co , - + - =1.
n
P
q
n=l
1-nuclear
operators
are
just
the
nuclear
operators.
3. EIGENVALUES OF NUCLEAR OPERATORS ON HILBERT SPACES. Since we will be concerned with eigenvalues, we assume all our spaces to be complex Banach spaces. Let H be a Hilbert space and let T € N(H) = N(H,H) have eigenvalues (A (T)) (ordered by decreasing modulus n
and counting multiplicities). The classical Weyl inequality [W] asserts that
so
the
eigenvalues
of
an
arbitrary
nuclear
operator
are
absolutely
convergent. Surprisingly, Hilbert space is the only Banach space enjoying this property:
42
Retherford: Hilbert space
Theorem [JKMR]. Let X be a Banach space. The following are equivalent: (a) X is isomorphic to a Hilbert space; (b) each nuclear operator on X has absolutely summing eigenvalues; and
(c) N(X) = lyiyx). Since
(c) is critical
to our work we discuss
it
in some
detail. Firstly, the notation T € IT -IT (X) means there is a Banach space Y and operators U € iyx,Y), V € Secondly,
if T e
Y,X) with T = VU. IT -IT (X)
then
T
has
absolutely
summing
eigenvalues. To see this, following Pietsch [Pi5] we say that A and B are related operators if
for Banach spaces X and Y. The importance of this notion is that
AB
and
BA
have, in
this case, the same eigenvalues. Thus if
T e TI *II (X) we have for some
Banach
II (X,Y),
space
Y,
T
=
BA
and
A
€
B
€
II (Y,X).
By
Grothendieck-Pietsch factorization we thus have the following diagram:
* X
Now
Uiai/32)(J2a2^i)€
= 5P (L (fi ))
and so this operator
has, by the Weyl inequality, absolutely summing eigenvalues. Clearly this operator is a relative of BA, i.e. T has absolutely summing eigenvalus. Our strategy is thus the following: to show that a Banach space X is isomorphic to a Hilbert space we need only show that for Te N(X), the eigenvalues of T, U
(T)), satisfy
Retherford: Hilbert space
43
00
I IVT)I
< +M
-
n=l
To accomplish this we show, under various hypotheses, that T (or T ) is in
II • II . 2
2
To do this
we need the "big picture" provided by a famous
result of Grothendieck.
4. GROTHENDIECK FACTORIZATION. As usual we denote by £ , 1 ^ p < +oo the Banach space of P 00
II All = V IA |p " "p [ L ' n1 J
sequences A=(A ) satisfying n
< +oo, and by I
the space
oo
of sequences A = (A ) with IIAII = sup |A I1 < +oo. M n
" "oo
Finally c
^ I n
n
denotes the closed subspace of I
0
those sequences A = ( A ) w i t h l i m A = 0 . n
n-X»
m
consisting of
oo
n
Let f®y, f€ X , y€ Y denote the rank one operator x
f(x)y.
Thus any non-zero T€ N(X) can be written
T =
V f ® y L
and f
* 0, y n
n
with n
Y
l l f II lly II
0, (a ) € c
+co
" n"
1
fn
1/2
f
n
TT^-TT ® a
yn
-n—n-
y
_
= £ g
n n
® z
n
and
n
This yields the Grothendieck factorization theorem for nuclear operators
44
Retherford: Hilbert space
[Gl]: if T€ N(X,Y), then T admits the following factorization:
where K
and K
are compact and 5 is a diagonal nuclear operator. Indeed 00
let O ), (g ), (z ) n
n
be as above and let K (x) = (g (x),K ((*•))= V ? z 1
n
n
2
n
L
n n
and 5((? )) = (|3 £ ). n
where 6
n n
Observe that 6 admits a further factorization
and 6 1
are the diagonal mappings defined by (|3
show from the definition that 6
2
^
€ TT (c ,1 ) (even N (c tt )) and 8 1
7r (£ ,1 ).
). It is easy to
n
2
2
0
2
2
0
2
Or, if one desires, a deep result of Grothendieck
€ 2
[Gl] also
oo* 2
covers the above situation. This diagram is now produced in it's entirety. Our strategy outlined above reduces to the intellectual (?) game -chasing the diagram! Grothendieck Factorization for
K ,K ,6,6 ,6
have the meanings as above.
T€ N(X)
45
Retherford: Hilbert space
Main o b s e r v a t i o n
(Related
operators): ^
(6 K ) ( K 5
1122
)
has
the
same
eigenvalues as T. Actually, starting anywhere in the
diagram and going all the
way around yields an operator with eigenvalues the same as T. For example, K K 5
is an operator on
c
with eigenvalues the same as T.
5. APPLICATIONS. In what
follows
we
will
operators as needed. We will write
introduce
X € H
classes
to mean
of
spaces
X is isomorphic to a
Hilbert space.
I. Theorem of Cohen-Kwapien [C], [Kl: An operator T e D (X,Y) provided T Theorem: Let
X
€ II (Y ,X ).
be a Banach space. The following are equivalent:
(i)
X € H;
(ii)
iyx.Y) c D2(X,Y) for every Banach space Y;
(iii) D (Y,X) c n (Y,X) for every Banach space Y; and (iv)
Same as (i) or (iii) with Y = I .
Proof: ( i M i i ) . If
X
is isomorphic to a Hilbert space
T € II (H,Y), by Grothendieck-Pietsch factorization
-» Y
C(K)
and
H
we have
and
46 and
Retherford: Hilbert space j a € ^ (H,H ) 2 0
so
a*j*
€ 5P ( H , H ) 20
and so
T % II ( Y * , H ) . 2 m #
(i)^(iii) is very similar: If Te D2(Y,H) then T € iyH,Y ) so by what was Just shown T**€ n (Y**,H) so T€ n (Y,H) since T** extends T with Y 2 2 ** canonically imbedded in (ii )=>( i)
and
Y
(iii )=>( i):
factorization.
If
(ii)
Let
T€
N(X)
and
consider
Grothendieck
holds
SK € TI(X,£) c D ( X , £ ) . Also S € 11 2 2 2 2 2 D (£ yl ) so T € D - D (X). Thus T*€ II -TI (X*) and thus has absolutely 221 22 2 2 # converging eigenvalues. But, the eigenvalues of T
are exactly those of T,
so X € H. If
(iii)
holds
KSe 2 2
DU.X) 2 2
c
II(£,X) 2 2
and
so
(K 5 )(5 K )€ll -II (X) and T has absolutely converging eigenvalues. (iv)=»(i) is clear since
all we used above was the space I .
An isometric
version of I is valid. Indeed isometric
are valid for most of
the
results
theorems we give. To state such results
requires conditions on the various norms of the ideals involved. Since we have not
explicitly given these norms we will not give the isometric
II. Theorem of Lindenstrauss-Pelczynski [LP]: results. One of the remarkable results of Grothendieck is the fact that £(L , H), the space of all bounded linear operators from an L (^)-space to a Hilbert space, coincides with
IT (L ,H). This result is proved using
what is now called the Grothendieck inequality [G2], [LP].
Theorem. If
X has an unconditional basis and £(X,Y) = II (X,Y) then X is
isomorphic to Proof:
< a
e.g.
n II
H
n
n
( x ) f ( x ) | = K < +00. Let n
i n Y,
n
'
||y || = a " n"
with n
X
and assume ||x|| = l,
( a )€ c , 0 < a n
sup
0
" " n
2 | g ( y ) | ^ 2.
ll ll
[MR] for such a c o n s t r u c t i o n ) . First,
N
is isomorphic to a Hilbert space.
n
let
n
T x =
Y
sup I | f II II llxll =11 f 11=1 and c h o o s e (y )
" n"
n+l
and
Let (x ,f ) be an unconditional basis for
||f || s M. Let a
I
V ^ f ( x ) y . Then Lt
i
i
i
(/3 ) € £ , n
2
|(J3 ) || "
n "2
= 1. For each N £ 1,
let
< 1, (see
Retherford: Hilbert space
47
n
||T x|| *
sup
V |f(x)||
M|x||(Z|e | 2 ) 1 / 2 ( s u p l | g ( y ) | 2 ) 1 / 2 s 2M|x| 1
so each T
HglUl
'
is continuous. By hypothesis there is a constant c, such that N
N
N
[ |T N (f i (x)x i )| = [ 1^: n=l
i=l
i=l N
* c sup
j
| f ( x ) f ( x ) | ^ c K | x ||.
II f 11=1
n=l
Since (/3 )
was arbitrarily chosen from
the unit
ball of I , (f (x))€l .
n
2
Knowing (f (x))e I for x € X, allows us without the sequence (/3 ). Since
n
doesn't matter) and ||y | = a
2
to repeat the construction above
(a )€ c
n
i
(being positive
and monotone
0
we now obtain (f (x))€ t . Since ||x | = 1 it
follows that X is isomorphic to t . Now let
T € N(Y)
and go to the diagrams. By hypothesis
(since X is isomorphic to I ) K € n {I ,Y) c n (£ ,Y). 2
2 1
1 1
Thus
so T has absolutely summing eigenvalues. A word of explanation is Grothendieck
in order. While our proof avoids the
inequality, the construction
theorem [Dl]. Moreover the
used requires the Dvoretzky
proof of the eigenvalue result we are using
requires the deep result of Lindenstrauss-Tzafriri
[LT] that a space is
isomorphic
each
to
a
Hilbert
space
if
and
subspaces is complemented. This result in So
we've
replaced
one
deep
only
if
of
its
closed
turn uses[D2]. result
with,
essentially,
equally deep results! Still, diagram chasing's fun. Let's continue.
two
48
Retherford: Hilbert space
III. Theorem of Morre11-Retherford [MR]: We operators
write
T € T (X,Y) if there
is a Hilbert
space
H and
A € £(X,H), B € £(H,Y) such that T = BA, i.e., T factors
through a Hilbert space. A Banach space
X
is in the class
2)
if there is a constant
00
such that for each
n
there is a subspace X
of X and an isomorphism
n
n : X n
n
>£
(complex n-space with the sup-norm) such that
KiiO Theorem: The following
are equivalent for a Banach space X :
(i)
X € H ;
(ii)
m , Y ) c T (X,Y) for any Y€ 2) ; 2
oo
(iii) m , c Q ) c r 2 (X,c Q ). Proof: (I )=>( ii ) =»( iii) is trivial (as is it all). we have by hypothesis that K (iii)=*(i). Diagraming again, we there is a Hilbert space
Then
H
€ T (X,C Q ) S O
with
AK SB € N(H) and as before X € H. This shows we may remove the "p" in the title of [MR].
IV. Theorem of Gordon-Lewis-Retherford [GLR]: Here C(X,Y) denotes the closure of the finite rank operators from X to Y.
Theorem: The following are equivalent: (i)
X€ H;
(ii)
C(X,Y) c r (X,Y) for all Y;
(iii) C(X,c ) c r (X,c ); 0 2 0 (iv)
C(Y,X) c T (Y,X)
(v)
C U ,X) c r (£ ,X).
for all Y; and
Retherford: Hilbert space
49
Proof. That ( i )=*( ii M iii ) and (i )=>( iv)=>(v) is trivial. i). Let T € N(X) and proceed to the diagram. By hypothesis Ke
T (X,c ) so there are operators
A, B
to and from a Hilbert space H,
with K = BA. Now 6 B € if (H,£ ) and so also to D (H, I ), i. e. T € D -D (X). 2 2 2 2 2 2 2 (iv)=>(i). Again let Te N(X). Now K ^ r il ,X) and thus A,B,H exist with K
= BA factoring through H. Hence
A I ( D is an isometric imbedding of Y into a suitable I -space (the index set T may be 00
uncountable) then
IT € N {X,l ( D ) . p
00
Theorem: The following are equivalent: (i)
X € H;
(ii)
T € N (X,£ ) => T*€ N (I ,X*); 2 2 2 2 # #
(iii) T € N {X,l ) =» T € QN [I ,X ); (iv) (v)
T € QN (X,l ) =» T € N (£ ,X ); T € QN (X,£ ) =» T*€ QN (£ ,X*). 22 22
Proof: C l e a r l y for
Hilbert
N c QN c II
regardless
of t h e s p a c e s
i n v o l v e d and s i n c e
s pK a c e s H , H , n ( H , H ) = y ( H , H ) , a g6 l a n c e 1
representation shows that
2
2
1
2
2
1
2
a t t h e Schmidt
50
Retherford: Hilbert space
Thus,
y (H ,H ) = N (H ,H ) = QN (H ,H ) = II (H ,H ) 2
for Hilbert
1
2
2
1
2
2
1
2
2
spaces. So we certainly have
1
2
(i )=»( ii )=»( iii)
and
(i)=>(iv)=»
(v)=»(iii). Thus we need only show (iii) =» (i). Let T € N(X). Rushing once again to the diagram we (iii)
K*6*
€
see that by construction, 8 € N (c ,1 ) so by
QN 2 U 2 ,X*)c
TI U ^ X * ) .
Also
S*K*€
iyx*,£ 2 ),
i.e.
T €ll -TI (X ) so T has absolutely converging eigenvalues.
VI. Theorem of Rosenberger [R]: We end these Hilbert space characterizations with a result which requires a generalization of the Hilbert-Schmidt
operators to a
class of operators different from the absolutely two summing operators. For
T € £(X,Y)
let
a (T) = inf {|T-A|: rank A * n-1}.
On
Hilbert space
a (T) = X (/TT n
) [Pi4].
n
Let, f o r 1 s p < +oo,
X,Y) = s (X,Y) = IT I T € m,Y)
| £ C£(TX+CO|
Then clearly for Hilbert spaces H , H
S (H ,H ) = y (H ,H ). 2 However, S (X,Y).
1 2
2
1 2
for arbitrary Banach spaces X,Y it is not true that II2(X,Y) = These
properties of
ideas a (T)
are
due
to
Pietsch
[Pi4].
Using
multiplicative
it is easy to show that
S • S c S. 2 2 1 Also
[JKMR]
if T
€
S (X) then
the
eigenvalues
of
T
are
absolutely
Retherford: Hilbert space
51
summable. Thus (same idea as the strategy) to show that X € H i t i s enough to show that
T € N(X) implies
(under various hypotheses)
that
T € S•
S2(X).
Theorem: The following are
equivalent:
(i)
X €H;
(ii)
f o r a l l p , 1 s p < co, N ( X , £ ) c S ( X , i ) a n d N ( X * , £ ) c S ( X * , £ ) ; p
(iii)
there
is
a
p,
2
^
2
p
(ii). It is easily seen that N (H ,H ) c II (H ,H ) for Hilbert p
spaces H ,H . A. Pelczynski
1
2
p
1 2
[P] has shown t h a t in t h i s case
n (H ,H ) = n (H ,H ) p
for a l l
1
2
2
1
2
p, l^p (x +[x ] n
n k
)
n' i=1 k=l i
every (n) = (n )°°_ u (n')°°_
is M-basic for
, (n )n(n')=0.
Then we say that (x ) is numerably strong M-basic if there n
exists a numerable sequence of partitions of (n)
Terenzi: M-basic sequences N
such
59
,=
that (x n^
+ [ x , ]°° )°° n t j j . i k=l
is
M-basic for
i= l,2,
Next theorem shows the existence of this intermediate type of M-bases.
Theorem II. Let
(x )
be a sequence of
B
with
[x ]
n
dimension and let
of infinite
n
(N ) be a numerable sequence of partitions of
N = (n ) i
ik k=l
u (n' )
ik k=l
(n)
, (n ) n (n' )=0 , i = l,2,... ik
ik
Then there exists (y ) of [x ] such that n
n
(i) (y ) is norming M-basis of [x ]' with (y ) n
n
for every m, where
n n—m
c span(x )
n n—q
m
(q ) is a non decreasing sequence of integers with m
q
>+oo; m
(ii) (y ) is numerably strong M-basic along the sequence n
i
of partitions of (n). Proof. By th. I of [5] we can suppose (x ) norming M-basic. n
Consider i=l. We have (x ) = x x
1
u (x
)_
n kt Ik " '1
u (x
)
n kt n' Ik k^t1
Set
y = x 1
i
, x
in
= x
n
;k
for k £ t ' , ;k oo
"lk
~
1
"lk
m=t 1
suppose that
(x
+ X') In
(N )
1 n=l
i s an M-basis of X 1
, with (x + X \ F ) In
1
In
60
Terenzi: M-basic sequences biorthogonal, (f
)c B
so that f (x) = F F (x+X') for forevery x of B
In
In
In In
1
and for every n. Consider f . n There exists an increasing sequence such
f (x
that
11
X
i,i
)* 0
for every
k
(p(l,l,k)) of integers
(otherwise
it would
be
n lp(l,l,k)
cf
111
), then we set
x
= x ln
n
lk
ik
for t £ k s p(l,l,l); *
moreover, for every k and for p(l,l,k)+l ^ i ^ p(l,l,k+l),
fi (x ) 11
n li
= X
X In
n li
li
77 / f (X 11
Then f (x' 11
ln
X n lp(l,l,k)
n
) = 0 for k > p(l,l,l); that is
ik
n
f
+ X'1
|x'
m = p(l,l,l) + lL
Consider
> )
.
If
(]
ln
s i x + X' I
'L^
lk
x'
m£p(l,l,l) •-
+ "lk
L I"
X' -"k^m
cF .
1J^2
j. F
U±
there
exists
increasing sequence (p(l,2,k)) _ of integers such that
f (x* n
* 0 for every k, hence we set
x
= x' In
lk
an
-1-
for p(1,1,1)+l < k ^ p(l,2,l);
1n lk
moreover, for every k and for p(l,2,k)+l ^ i ^ p(l,2,k+l),
)
ipd,2,k)
Terenzi: M-basic sequences f „ x
(x* 12
>
=
In l i li
x In
12
li
n lp(l,2,k)
such that
= span (x )
In n>l
;
n n>l
c span (x )
I n n^m
l
In n>l
span (x ) (x )
\ )
l n
So proceeding we get y u (x ) 1
)
In
7Z / * f (x
In l i
61
for every m, where (q ) i s a non
n n^q lm
ltn
decreasing sequence of integers with q — > +oo; lm
(x In + [xIn' ]°° )°° j=t* k=t Ik
lj
1
is
M-basic.
1
Consider i=2. We have
(X
ln)n>2=
(X
2
1,2
; x
2n' 2k
)
Ik
= Set y =x
ln
(X
=x
In' 2k
k ^ s U( X l n ' 1
Ik
.n2k K±e2U
(X
for k £ s
W
1
ln'2k W 2
2
Suppose that 00
x' = [x
]
ln
(x 2n
(f
2k^2
+ X')°° 2 n=l
p
, x = n 2
m=
x
S2Lln2k
21
2J
J = S 2
Jk^
M-basis of X with (x +X',F )°° biorthogonal, 2
2n
and for every n.
We have that
ln
2n
2
2n n=l
)°° c B* so that f (x) = F (x + X' ) for every x of B
2n n=l
Consider f .
-i
i 00
+ [x
2n
2
62
Terenzi: M-basic sequences (n
= (n(1)) ^ (i) U (n(2)) > (2) with
)
2k k^s
2k
2
k^s
2k
2
k^s
2
Consider
two
increasing
sequences
2
1
of
integers
(p
(2,l,k))°°_
(i =l,2)
so
that
f
21
(x (i) In
(i) 2p (2,l,k)
) * 0 for every k and for i=l,2
(maybe that one of these two sequences of integers does not exist, for example for i=l, then we shall set x
(1) = x
2n
ln 2k
(1) for k^s
and we
2
2X
shall consider only i=2). Now we set, for i=l,2,
x
s (l) s k s p (i) (2,l,l);
(i) = x (i) for 2n
2k
moreover, for every k and for p U) (2,l,k) + 1 ^ j s p ( i ) (2, 1, k+1),
f (8)
Then
X* (i) = X (i) n 2j °2j
f
(x*
) =
0
for
21
(X (1)) In X
f
21
large
(X (i) ) In 2p(2,l,k)
k;
moreover
(i) 2p(2,l,k)
n
(x*
+
[x' , ]°°_ t) ^
still M-basic because by (7) and (8) we have that
[*' . ] " _ .
(x* 2n
)^
= Ix , 1". . ;
c span(x
k^m
)^ In
for every m, where (q(l,m))
k2:q(l,m)
is a non decreasing sequence of integers with q(l,m)—> +00.
In the same way we consider f
and so on.
is
Terenzi: M-basic sequences
63
So proceeding we get (y , y ) u (x &
&
1^2
span (x
)
)
= span (x
2n n>2
(x
)
such that
2n n>2
)
;
In n>2
c span (x
2n n^m
)
for every m > 2, where (q
In n^q
) is a
2m
non decreasing sequence of integers with q —> +« ; 2m
(x
2n
+ [x Ik
(x
2n' lj
+ [x
^3
)°°
i s M-basic;
j=s* k=s 1 1
, ]°° ,) i s 2n
Consider i=3. We set y = x
]°°
J=S
2J
M-basic.
2
; then we have 2,3
= (x2n )k^r U (x2n )k^r' Ik
= (X *• x «
~
Set x
3n' 3k
= x
2n* 3k
Ik
U (X
'. «^
t
1
)
U IX
J
for k ^ r . Suppose that K 3
X' = [X 3
]
2n' k^r' 3k 3
(x + X')°° 3m
(f
1
)
^
. X = 3
M-basis of X
3 n=l
)°°
f|
'' m=r
tx
c B*
so that f
Consider f . 31
We have that
"
in
2n' 3j
(x) = F
]
j=r'
+ X' , F
3n
3n
and for every n.
tx
3k
with (x
3
3n n = l
+
2n
3
] 3
k^m
) biorthogonal,
3n
(x+Xl) for every x of B 3n
3
64
Terenzi: M-basic sequences
~r3
3k
~P3
3k k ^
k2:r3
(3)
~ ri
Ik k^rj
_
~P2
2k
k ^
,
P
F
~ 3
P
~ 3
~F2
~ 1
We consider again four sequences of integers (p
(3,l,k))°°_
(1=1,2,3,4) so that
f
31
(x
2n
(i) ) * 0 for (i) 3p (3,1,k)
e v e r y k and f o r
1=1,...,k
.
If some of these sequences of integers do not exist, then we shall proceed as for (p(i)(2,l,k)). It is now sufficient to set , for i=l,...,4,
x
r(i)-s k ± p(i)(3.1,l) ;
(i) = x (i) for
3n 3k
2n 3k
3
moreover, for every k and for (p(l}(3, 1, k) + l ^ j * p (l) (3,1,k+1),
f (x (D) 31
2n X
2n (U (i, 3
3p
Now the procedure is clear, for (f ^
)
(
(3,l,k)
and for i>3.
3n n>l
So proceeding we get (y ) as in the thesis (indeed, since (x ) is norming and since (y )
n n^m
n
n
c span (x )
n n^q
with q —-> +oo, (y ) is norming too). This m
m
n
completes the proof of th. II. We shall say that an M-basic sequence
(x ) is quasi-strong n
M-basic if (x ) has the possibility to become strong M-basic by means of n
the procedure of Th. II; that is if, after removing the not regular points along a numerable sequence of partitions of
(n), the sequence becomes
Terenzi: M-basic sequences
65
strong M-basic. About
the stability we point
out
that
a middle
strong) (quasi-strong) (strong) M-basic sequence keeps its sufficiently
"near" sequences:
it is sufficient
(numerably
properties for
to prove this for the
strong M-bases.
Proposition.
If
(x ) is strong M-basic in B,
there exists
(e ) of
n
n
positive numbers such that every (y ) of B with fly - x II < c n
n"
n
for every n
n is strong M-basic. Proof. If (x ,f ) is biorthogonal suppose that n
n
= l/(2 n + 1 ||f ||) for every n.
||y - x II < e 11
n"
n
" n"
n
Following the techniques of the proof of the Krein-Milman-Rutman theorem (see [2] p. 84-99) it immediately follows, for every (a ) m
, of numbers,
n n=l
H I Ia * h I Ta y I s I I Ta * 2 £
L< n=l
n
n
IIn=l L. n
n II
^
In=1 L n n
If (y ) is not strong M-basic there exist (n) = (n ) u (n*), (n ) n (n')= 0 n
°
k
k
k
k
and y of B so that
p
r
_p
(10)
II y + iy niJ III = i ; III y - fI L) ap, kyn:,k + Y } II < ?L0
s.t.
a < 1
Take now 2 a-l elements of the basis (e ), le1, e1, . . . , e1 n \ 1 2 Pa J 2
1A ej| < a , (1 = 1
2 ^l) > 0)
Perform
in
[e,e,e,...,e 1 1 2
-1
(this is always possible since ||A e |
the
], given by p
I such that
change
of
o.n.b.
the
subspace
Rod6s: Semi-Fredholm operators M :(e . e 1 ^ 1 , . . . ^ 1 1 1 2* p v
1
79
)
> (u , 1'
P
1 2-1
u
) P
2
1
- Clearly 1^=2
( e ^ e*±e*±...±e
) and then Vie B(l), 1
-(P/2)
-(p /2) l
k. n"
L e t c > 0 , 3 p e IN s . t .
2
*>Mn=l
Perform t h e change o f ( e ; n € IN) n
o.n.b.
> ( v ; n€ .IN), n
construct iong (v ; n€ IN) by the following blocks: n ( B . 1 ) M: ( e . e . e P
l
.. . . , e
k+i
(B.2)M:(e,e p
2
)
e ) k+2-2p-2
k+2 p
> ( v , . . . , v _
k+2p-l
k+2
(B.k) M : (e , . . . , e ) k p P k . 2
1
> (v 2P+1
> (v (k-l)-2p+l
Then -(p/2)
|Av I = 2
||Ae ||
ii
»
i ii
Vi € (B. 1)
i "
-(p/2) IIAv || = 2 II
|
II
||Ae || "
flAvJ| = 0 Vi>k2
k" p
2
Vi € ( B . k )
P
) .
,v ) 22P
v ) k-2p
P and v =e Vn > K-2 n
n
Rod6s: Semi-Fredholm operators
83
Taking into account
||A|| > | | A e J > | | A e 2 | > . . . >
|AeJ,
-(p/2)
I ||AV II < n " n" n€(N
\ I ||A| 2 ^ n » "
=2
|A|| ) - < e. " " / n n= 1
n=l
Since
( ||Av ||; n€lN) i s n o n i n c r e a s i n g
, ) (Q,A)< e where Q i s Lt
n
the Hilbert cube associated to (v ; n€ IN) and inf jy(Q,A);Q Hilbert cubeU = 0. Case 2. Codim ker A=0. Let e>0. Take a sequence of positive real numbers (c ; neIN) n
s.t. y c < G. L n Let •
(ker A)
(u ;n € IN) and (e ; n€ DM) be o.n. bases of ker A and n
n
respectively (B. 1 ) . F o r c >0
3 p € IN s . t .
- | ± M||x||
, Vy% B(F*),
which proves that T is continuous and ||T|^M. From the definition we get *
*
•
*
T (y ) = (T (y )) and reasoning as in the first part of the proof we n
obtain (IV). Let us consider now some special classes of operators. For a pair of Banach spaces E and F we shall write: - K(E,F)={Te L(E,F): T(B(E)) is relatively compact} (compact operators). - w(E,F)={T€ L(E,F): T(B(E)) is weakly relatively compact} (weakly compact operators). -
D(E,F)={T€
L(E,F):
T
sends
weakly
Cauchy
sequences
into
weakly
convergent sequences} (Dieudonne operators). - DP(E,F)={T€ L(E,F): T sends weakly Cauchy sequences into norm convergent ones} (Dunford-Pettis operators). -
U{E,F)
=
{
T
€
L(E,F)
unconditionally
: T
sends
convergent
ones
weakly
Cauchy
sequences
} (unconditionally
into
converging
operators). - S(E,F)={TF belongs to some
n p
operator ideal &, clearly all the operators in the representing sequence
98
Bombal: Vector sequence spaces
(T ) of T belong to the same operator ideal. The converse is not true in n
general, as the following examples show.
Example 1.2. For each n€ IN, let us take a Banach space E *{0} of finite n
dimension, and let I be the identity operator on (ZeE ) . Then, each I n p
is
n
a finite range operator, and so it belongs to all the classes of operators considered, but: a)
I
is
an
isomorphism.
Hence,
it
is
neither
compact
nor
strictly
singular. b) When p=l,
(SeE )
contains a copy of I , and so
n 1
compact. c) When
I is not
weakly
1
p=0,
(Z©E ) contains a copy of c , and so I is not n 0 0 unconditionally converging. d) When p>l, (ZeE ) is reflexive. Hence, I is not Dunford-Pettis, because n p
any Dunford-Pettis operator on a
space that contains no copy of I , is
compact by Rosenthal's I -theorem, (see, f.i [5] 2.e.5). In order to obtain some results in the positive direction, we need the following lemmas: Lemma 1.3. Let E (neIN), F be Banach spaces, l^p 0, there exist
n e (N such that
Sup £ * ||II (x)| p < c X€A n^n Then
lim sup 1 V T n (x) - T(x)| = 0 mX
X€A. " L
n n
n=l
Proof. Let e>0 and choose m
Sup
such that
I ||n (x)| p s (e/||T||)
X€A n^m
n
"
Bombal: Vector sequence spaces Then, if y e B(F ) and m>m
99
for every X€A we have
][ T II (x)-T(x), y* >| = | £ | n>m n=l
f I llV
re»
^n>m * * by (IV) of th. 1.1 the result follows, taking the supremum when y €B(F ). We also need the following well known result:
Lemma 1.4. ([2], Lemma 6). Let H be a subset of a Banach space. If for every e>0
there exists a (weakly) compact subset H
of E such that
H c H +cB(E), then H is relatively (weakly) compact.
Theorem 1.5. Let E
(n€lN), F be Banach spaces, l^pl , T is weakly compact, if and only if so is each T . n
Proof. a) Let us consider first the case of unconditionally converging operators. k k k Suppose each T is unconditionally converging and let Ix (x =(x )€ E) be n n a weakly unconditionally Cauchy series in E. It suffices to show that ||T(x ) I tends to zero. As (T(x )) is weakly null, we only need to show that H={T(x ): k€(N} is norm relatively compact. In the first place, let us notice that the set H ={T (II (xk):k€(N> is relatively norm compact for each n
n
n
n€(N by hypothesis. Reasoning as in the case E =K for every n, one can n
easily prove that the set A={x : k€lN}, being relatively weakly compact, satisfies condition ( + ) of lemma 1.3. Then, given e>0, there exists meIN such that I E 11
*•*
n>m
T II (xk) II < e for every k, i.e. n
n
"
T(xk)€ H e + eB(F), for every ke IN,
100
Bombal: Vector sequence spaces
where H
is the norm closure of H+...+H , a norm compact set. Lemma 1.4
applies , giving the
result.
The proof
for
Dieudonne and Dunford-Pettis operators is
k
similar: let (x ) be weakly Cauchy in E. Since (T(xk)) is weakly Cauchy, to prove that it is weakly (resp. norm) convergent, we only need to show that
H = {T(x ): ke(N>
is relatively weak (resp. norm) compact, and this
can be proved as before. Let
us
remark
that
another
proof
for
the
case
of
Dunford-Pettis operators is included in [3], Prop. 14.3. b) See [3], prop.
14.4.
Corollary 1.6. Let & be an operator ideal. Then, with the notations of theorem 1.5, we have a) 0((Z@E ) ,F) c 0((ZeE ) ,F) if and only n 1
c i//(E , F ) for
every
if
n 1
ne IN (where
0(E ,F) c n
\p = DP, U or
D).
n
b)
For
Kp0 and, for every m, a q>m such that || V T -IT -T|| > e. Take q
and x*€ E
such that ||x II^1 and
> c.
There is a p >q such that
T -TT (x 1 )! > c. n
n
"
n = q +1
Take q >p such that II V T -IT -Til > c and proceed inductively. Then we get 2
1
La n
n
n=l
a sequence such that
(xn) in B(E) and a sequence of natural numbers q
f) = V^10/4 > (2+V2)/4 = e (u*: £ 2
2
2
>l* ). 00
As of yet, it is unsettled in general, if the entropy numbers of an operator and its dual are proportional, i.e. whether
108
Konig: Entropy numbers
(1)
3 l u is compact «=> e (u)—£) «=» e (u )—£). Even the more restricted and more accesible question whether
(2)
3 l^a,cY
known.
c^e, , (u):s e (u*)s cer ,(u) [akj k Lk/aJ
(2) would
imply
that
for
any
symmetric
sequence space E one has
As shown recently by Tomczak-Jaegerman [7], (3) is true for u: X
>Y if X
or Y is a Hilbert space. See also [5]. In some particular cases and examples positive partial answers are known. E.g. for identity maps
Id: ln—>£n p
the strongest statement (1)
q
is true, see Schiitt [6]. The proofs for general operators all rely (in various forms) on a reduction to the Hilbert spaces case. Recall that a Banach space X if of type 2 provided that m
3 lY*. Together with the fact that
similar
statement
J^Ti"1: H
>K
holds
for
has self-dual
entropy numbers (Hilbert space case), we find estimates of the form (2). The reduction to rank k maps is done using the approximation numbers of u, a (u) = inf «||u-u II I rank u < k>. Carl showed that for k ^" k" ' k J * u:X—>Y with X, Y of type 2, the geometric mean of the Gelfand numbers is proportional to the entropy numbers. The same statement holds for the geometric mean of the approximation numbers. The reduction to rank k maps uses that the approximation numbers of compact maps are self-dual. In general Banach spaces X and Y, there is another positive partial result of form (2) for finite rank maps
u: X
>Y
and k larger
than a multiple of the rank of u: Proposition 2 [3]. For any A>0 there is
a = a(A)
Banach spaces X,Y and any finite rank operator u: X one has
e
[ a k ] ( u ) *2\{u
]
•
e
[ak](u '
such that for any >Y and k ^ A-rank u
110
Konig: Entropy numbers
This follows from the following duality result for covering numbers:
Proposition 3 [3]. There is
1 < c < oo such that for all n e IN, all
compact, absolutely convex bodies K ,K r~1NfK° p K ° ) 1 / n < NfK
(A)
2
Here
K
, K
1
in
Rn and any
rK l 1 / n < rNfK° 1 2
2
e > 0
rK°)1/n 1
denote the polar bodies with respect to the euclidean norm
on Rn. To prove proposition 2 for rank u = n-dimensional spaces X and Y (to which it reduces easily), one has to apply proposition 3 to K *
= u(B v ), K 1
A
= B . 2
i
Then K° = u ^(B • ) , K° = B * 1
A
2
Y
The proof of proposition 3 uses volume arguments as well and uses an inverse form of the Brunn-Minkowski inequality due to Milman [4] for the reduction to Hilbert spaces. Let D denote the euclidean unit ball in Rn. Inverse Brunn-Minkowski-inequality [4]. There is 1 ^ c < oo such that for all compact, absolutely convex bodies K c Rn n
v:R
n
JR
there is a
linear map
with det v = 1 such that for any OO
[vol (v(K)+ eD)] 1 / n s c[vol (K)] 1/n + a e[vol (D)] 1/n n
n
n
n
with a —>1 for n—*». n
By the Brunn-Minkowski inequality, always (v(K)+ eD)]1/n £ [ v o l (K)]1/n + e[vol
[vol n
n
(D)]1/n n
holds. One can use the Brunn-Minkowski inequality and its inverse to give a more precise formula for covering numbers N(v(K),D) in terms of volumes. A special case of theorem 2 of [3] is Proposition 4. There is 1 ^ c < oo such that for all compact, absolutely convex bodies K c Rn all e>0
there is
v:Rn—>(Rn
with
det
v = 1
such that for
Konig: Entropy numbers
111
VOl (K) ,1/n
max VOl (K) n vol"
1 / 1 1
l n (D) J /C>1)' Thus the volume ratio determines the covering numbers. REFERENCES. [1]
Gordon, Y; Konig, H.; Schutt, C. (1987). Geometric and probabilistic estimates for entropy and approximation numbers of operators. J. Approx. Th. [2] Gordon, Y. ; Reisner, S. (1981). Some aspects of volume estimates to various parameters in Banach spaces. Proc. Workshop Banach Space Theory. Univ. Iowa, 23-53. [3] Konig, H. ; Milman, V. (1987). On the covering numbers of convex bodies. GAFA seminar Tel Aviv 1984/85, in: Lecture Notes in Math., Springer. [4] Milman, V. (1986). Inegalite de Brunn-Minkowski inverse et applications a la theorie locale des espaces normes. C.R.A.S. 302 25-28. [5] Pajor, A.; Tomczak-Jaegermann, N. (1985). Remarques sur les nombres d'entropie d'un operateur et son transpose. C.R.A.S. 301, 743-746. [6] Schutt, C. (1983). Entropy numbers of diagonal operators between sequence spaces. J. Approx. Th. 40, 121-128. [7] Tomczak-Jaegermann, N. (1987). Private communication.
MIXED SUMMING NORMS AND FINITE-DIMENSIONAL LORENTZ SPACES G.J.O. Jameson University of Lancaster Great Britain
INTRODUCTION This largely expository article describes some recent results concerning
the mixed
summing norm
II
as applied
to operators
on
p>i
I -spaces (especially the finite-dimensional spaces £ n ). The whole subject 00
00
of summing operators and their norms can be said to have started with a result on II
- the theorem of Orlicz (1933) that the identity operators
in I
are
and I
(2, 1)-summing.
However, since that time the study of
mixed summing norms has been somewhat neglected in favour of the elegant and powerful theory of the "unmixed" summing norms II . A breakthrough has p
now been provided by the theorem of Pisier
[7], which does for mixed
summing norms what the fundamental theorem of Pietsch does for unmixed ones
(see e.g.
[2]). One version of Pisier's theorem states that the
operator can be factorised through a Lorentz function space
L
(A). Such
p»i
spaces were introduced in [5], and are discussed in [1] and [4]. However, it is not easy to find a really simple outline of the definition and basic properties
of
these
spaces
adapted
to
the
(obviously
simpler)
finite-dimensional case, so the present paper includes a brief attempt to provide one. It is of particular interest to compare the value of II TI for operators on 2
I oo
or
and
£n. It is a well-known fact, underlying the oo
famous Grothendieck inequality, that there is a constant K, independent of n, such that for all operators T from
ln
to
KIITII. This equates to saying that II (T) s
I
K'll
or
I , we have II (T)^
(T) for such T. It is
natural to ask whether this relationship holds for all operators from £n 00
into an arbitrary Banach space Y. It was shown in [3] that this is not the case: for each n, there is an operator T
on ln such that
Jameson: Mixed summing norms n (T )* I (log n ) 1 / 2 n 2
n
2
113
(T ).
2,1 n
It was also shown that this example gives the correct order of growth: there exists C such that for all operators on £n, 00
n (T) s C (log n ) 1 / 2 IT (T),
(so the ratio of IT to TI
grows very slowly with n).
We shall describe a
different approach to both these results, using Lorentz spaces. In the case of the second result, this will lead to a more general statement, which is due to Montgomery-Smith (6).
Notation £n
We denote by [n] the set {l,2,...,n} and by
the space (Rn
00
(or C n ) with supremum norm. We denote the ith coordinate of an element x of R n by x(i),
and the ith unit vector by e . For an operator T from £ (S) i
oo
to another normed linear space Y, the summing norms
II (T) p
and
II (T) P,I
are defined by:
TUT) = s u p { [ [ | | T x i | | " ] 1 / P : \ \ l
IXJX
3
i -V 1 / p
in which all finite sequences of elements (x ,...,x ) are considered.
Ik
11
As usual, p' denotes the conjugate index to p: - + -, = 1.
Pisier's theorem Let T be an operator defined on
t (S) 00
Suppose that that
there
is a positive
linear
(for any on
functional
||
t>.
Clearly, d ^ t ) = p^ for x*(k+l) s t < x*(k). Now define
(1)
P 1 / p tx*(k)-x*(k+l)) k
By Abel summation, this is also equal to
(2).
Jameson: Mixed summing norms
117
If some of the x (k) coincide, then different
choices of the j
are
possible. However, it is clear from expressions (1) and (2) that this does not affect the definition. It is not quite so clear that || • ||
is a norm.
We return to this point below. First we list some properties that follow at once from the definition.
then \\y\\X
Proposition 2. (i) If |y|^|x|,
j
± |x||A ^
(ii) For A c [n], l l ^ l ^ = A(A) 1/p ( = j x ^ ). (iii) ||x||* i = |x|* for all x. (iv)
IIxII X
11
< p1/pl|x||
"p,l
n
(v) I f \< Kji,
"
(note
"
=X([n])). n
\\x\\X
then
p
"oo "p,i
± K 1 / p |x||* i "
"p,i
for
all
x.
Proof. Immediate.
in DRn,
Proposition 3.
For all
x
Proof. Write y
= x (k)-x (k+1), so that l/p
r k
E
,1/p
by Holder's inequality. But
Corollary 3. 1. L
(A). Then
p,l
Let II p,l
then n p, 1
(K ) = 1. n
K
(K ) = n
l/p'
V p y
= ||x||
and V y
= x (1) = |x| .
denote the identity operator from ||K || = A([n]) 1/p . " n"
l^
In particular, if X([n]
to =1,
118
Jameson: Mixed summing norms
Proof. This follows from the easy implication in Pisier's theorem, given that ||xI
£n
= y>(|x|), where
i
Proposition 4. Let q be any seminorm on |y| = xA, then
q(Y) s A(A) 1/p .
Then
(Rn
such that if
q(x) ^ |x||A
A
for all
A Q [n]
and
x.
p, 1
Proof. With notation as before, let f
= sgn x(j )e . Then
x = Y x*(k)f = V u , L
k
L k
where
uk = [x*(k)-x*(k+l)l(f [ J i +... By hypothesis, q(f +...+f ) ^ p 1/p , so
q(x) * I qCuJ k
1/p rx*(k)-x*(k+nip^ "k
_
k
= Ml* • " 1 11
p,
Corollary 4.1.
A ||x| ^ ii i i p
We
mention
L
A ||x| " "p,i
briefly
for all x.
some
further
basic
facts
relating
to
(A) which can be proved easily with the help of Proposition 4. Except
for the fact that || • | »
further development.
"p,i
is a norm, these results are not needed for our
Jameson: Mixed summing norms Proposition 5.
Let (r
r ) be any permutation 1
=
xy, the dual norm to
|y|J.fW= sup |x(A)" 1/p J |y|: A C [n]|. (Apply Proposition 4 with
Proposition 8.
q(x)=|<x,y>|.)
Given positive elements
x ,...,x 1
of L
where \\-\\ stands for | • fl . In other words,
(A), we have
p,l
k
L
(A) satisfies a "lower
p-estimate" with constant 1. (One proves that the dual satisfies an upper p'-estimate.)
For p 2: q ^ 1, one defines llxll 11
to be
"p,q
l/q
At the cost of some extra complexity, the above results can be adapted for this case.
120
Jameson: Mixed summing norms The second form of Pisier's theorem
Theorem 9. Let
T
be any operator on
ln. Then there is a probability 00
measure
X
on
[n]
such that
||TX| , P 1/p n pi (T) |x|J t l L U ) {so that n (K ) = 1) and IIT | * p 1 / p n (( T ) . P . I all x. Hence T p , =l nT K , where" 1 K " for is thep , lidentity operator from in to n
In
oo
P r o o f . L e t (p b e t h e f u n c t i o n a l g i v e n b y T h e o r e m 1. T h e r e e x i s t X ^ O s u c h that V X Li
= 1 a n d K such that P(x) = II(x,x), where
II is a bilinear functional defined on ExE. Let us recall that the bilinear functional IT associated to P in the above sense is unique if it is symmetric and that P is cotinuous if and only if the same is IT, or if and only if it has a finite norm
|P|| = sup ||P(x)|: |x| = lj = sup j|P(x)|-||x||"2:x*oj It is an old open problem the characterization of the normed linear
spaces
of
dimension
^3
satisfying
the
property
that
every
continuous 2-polynomial defined on a linear subspace of it can be extended to the whole space preserving the norm. In Aron attribute to R. M.
and
Berner
(1978) appears
an
example,
that
they
Schottenloher, of a normed linear space in which not
every continuous 2-polynomial defined in a linear subspace of it can be extended to the whole space preserving the norm. On the other hand, not for 2-polynomials functionals, it is conjectured
but for bilinear
in Hayden (1967) that the inner product
126
Benftez & Otero: 2-polynomials
spaces are the only normed linear spaces satisfying and analogous property of extension preserving the norm. In this paper we give an example (as far as we know the first to be published) of a simple family of real normed
linear spaces of
dimension 3, whose norm is not induced by an inner product and such that every 2-polynomial defined in a linear subspace of it can be extended to the whole space preserving the norm. In a certain sense our example is intermediate between the negative example of Schottenloher and the well known positive example of the
inner product
spaces, since
in the most simple case of the real
3-dimensional spaces the unit ball of a inner product space is defined by a quadric, in the example of Schottenloher such ball is defined by the intersection of three quadrics and in our example it is defined by the intersection of two quadrics.
Firstly
we
summarize
for
the
real
case
the
example
of
Schottenloher. With the same arguments used in Aron & Berner (1978) for the space C
it is not difficult to see that in the space E=R
with the norm
||(x,y,z)|| =
P(x, y,z)=y +yz+z
sup(|x|, |y|, |z|),
the 2-polynomial
endowed
defined by
cannot be extended from the linear subspace L={(x,y,z):
x+y+z=O} to the whole space E preserving the norm, since ||P| =1 and for every extension P of P to E we have that |P|| >1. 2
2
In geometrical terms this means that the ellipse y +yz+z =1, x+y+z=O is around the cube S={(x,y,z):sup(|x|,|y|,|z|)=1> but such ellipse is not the section by the plane x+y+z=O of any quadric circumscribed to the mentioned cube. We shall need for our example a previous lemma 2 Lemma 1. If A and B are 2-polynomial in IR such that 0 * sup [A(x),B(x)], (x€ R2) then there exists O^t^l such that
Benitez & Otero: 2-polynomials 0 s tA(x) + ( l - t ) B ( x ) ,
127
( x € IR 2 ).
Proof. It is obvious when either A(x) of B(x) are non negative for every x€ IR2. In the other case let y and z be such that
A(y) < 0 ^ B(y), B(z)0
Then
from
the
expansion
of
[tA+(1-t)B](ru+sv)
we
obtain
that the wanted t is such that
ta(u,v) + (l-t)|3(u,v) = 0.
Corollary. If P, A, B are 2-polynomials in IR such that P(x) * sup [A(x),B(x)], (x€lR2) then there exists O^t^l such that P(x)s tA(x)+(l-t)B(x),
(xe IR2).
Proposition 1. Let E be the linear space IR endowed with a norm whose unit sphere is given by S = {x€ IR3: sup[A(x),B(x)] = l>
128 where
Benitez & Otero: 2-polynomials A
and B
are positive
semidefinite 2-polynomials.
Then
every
2-polynomial defined in a linear subspace of E can be extended to E preserving the norm. Proof. We shall consider only the non trivial case in which P is a non zero 2-polynomial defined in a two dimensional linear subspace
L = ker
Benitez & Otero: 2-polynomials
129
Then y ^ y ^ S, PCy^X), P(y2)y ) s P2(y.) sup[A(y),B(y)], (y e L), (i = l,2)
where the first inequality follows from the fact that P is indefinite and the second is true since in other case there would exist y € S n L such that IT2(y ,y )> P2(y ) and hence i
o
i
for t>0 sufficiently small, in contradiction with the definition of y. . Let «// and 0
be norm-preserving extensions to E of the non
zero linear functionals TT(y ,.) and TT(y ,.) such that i// is proportional to t// when so are TT(y , . ) and TT(y , . ). Then
02(x) * P2(yi) sup [A(x),B(x)],
and if we take v € (ker 0
n ker \p )\L and define
we
(i=l,2), ( X G E )
(X€E)
obtain
* P2(y.) sup [A(x),B(x)], (XG E)
from which it follows that
|P(x)|
< sup
[Pty^,
= ||P||L sup
|P(y2)|]
[A(x),B(x)l
sup ,
[A(x),B(x)l
(x€
E).
130
Benitez & Otero: 2-polynomials
as we wish to prove.
SOME REMARKS. A classical and involved theorem due to Blaschke (1916) and 3
Kakutani
(1939),
says that the unit sphere S of a norm
in R
is an
ellipsoid if and only if every section of S by an homogeneous plane can be extended to a cylinder supporting (or circumscribed to )S. On the other hand the non less classical theorem of Hahn and Banach is a generalization or abstraction of the obvious fact that every straight
line supporting S (obviously
in an homogeneous plane) can be
extended to a plane also supporting S. Roughly intermediate
speaking
problem
which
we
have
has
an
treated
in
intermediate
this
paper
answer.
with
Namely
an the
hypothetic extension of conies supporting S in an homogeneous plane to quadrics also sopporting S. With
regard
to
this
problem
we
have
seen
that
the
above
extension is not always possible when S is a cube (Schottenloher example) and it is always possible when S is a "barrel" (our example) From this stimulating points of view (a little far, certainly, from the barreled spaces) we can state in easy terms many
interesting
questions about this old and seemingly difficult problem. Are the barrels the only three dimensional examples for which is valid a quadratic Hahn-Banach theorem?. What about cylindrical extensions of circumscribed conies?. What about either dimension >3, or 2n-polynomials, or non homogeneous polynomials?. What about extension of bilinear functionals?. Etc, etc.
THE COMPLEX CASE. As we have pointed out before the Schottenloher example was given originally in the complex space C . We translated it automatically to the space IR and we gave our example in this real space.
Benitez & Otero: 2-polynomials
131
In fact, as a corollary of the following elementary lemma we obtain that the problem of extension preserving the norm of continuous 2-polynomials in complex spaces, can be reduced in all the cases to the real case.
Lemma 2. Let E be a complex normed linear space, P: E
>C be a continuous
complex 2-polynomial in E and IT be the continuous symmetric real bilinear functional associated to the continuous real 2-polynomial P =Re P. Then P(x)=TMx, x)-ill (ix, x),
(x€X) and ]P||=[P| .
Proof. Let
P(x)=Pi(x) + iP2(x), with P ^ x ) , Pg(x) € (R
and let IT (resp. II , IT ) be the continuous symmetric complex (resp. real) bilinear functional associated to P (resp. P , P ). The first part of the lemma follows from the equalities
TT (ix,x) + iIT (ix,x) = TT(ix,x) = iTI(x,x) = -II (x,x) + iTI (x,x) 1 2 2 1 The second part is an inmediate consequence of the fact that for every x€ E there exists & € C such that |# 1=1 and P(# x)€ R. x
Proposition 2.
' x'
Let E be a complex normed linear space, L
subspace of E and P: L continuous real
> C
=
Re
P
can
be
extended
to
a real
the same norm, then the same is valid for the
original complex 2-polynomial P. Proof. If
= n (x,x) , (x€ E)
is the extension of P
a linear
a complex continuous 2-polynomial. If the
2-polynomial P
2-polynomial in E of
P
x
to E, then
P(x) = n (x,x)-iTI (ix,x), (x€ X)
132
Benitez & Otero: 2-polynomials
is the extension of P to E.
REFERENCES. [1] Aron, R.M. & Berner, P.D. (1978). A Hahn-Banach extension theorem for analytic mappings. Bull. Soc. Math. Franc. 106, 3-24. [2] Benitez C. & Otero, M. C. (1986). Approximation of convex sets by quadratic sets and extension of continuous 2-polynomials. Boll. Un. Mat. Ital. , Ser. VI, Vol. V-C N.I, 233-243. [3] Blaschke, W. (1916). Kreis un Kugel. (1956). Reprinted Walter de Gruyter Berlin. [4] Hayden, T.L. (1967a). The extension of bilinear functionals. Pacific J. Math. 22, 99-109. [5] Hayden, T.L. (1967b). A conjecture on the extension of bilinear functionals. Amer. Math. Monthly, 1108-1109. [6] Kakutani, S. (1939). Some Characterizations of Euclidean space. Japan Jour. Math. 16, 93-97. [7] Marinescu, G. (1949). The extension of bilinear functionals in general euclidean spaces. Acad. Rep. Pop. Romane. Bull. Sti. A. 1, 681-686. [8] Moraes, L.A. (1984). The Hahn-Banach extension theorem for some spaces of n-homogeneous polynomials. Functional Analysis: Surveys and Recent Results III. North-Holland, 265-274. [9] Nachbin,L. (1969). Topologies on Spaces of Holomorphic Functions. Springer Verlag.
ON SOME OPERATOR IDEALS DEFINED BY APPROXIMATION NUMBERS
Fernando Cobos Dpto. de Matematicas, Univ. Autonoma de Madrid, Madrid, Spain Ivam Resina Inst. de Matematica, Univ. Estadual de Campinas,S.Paulo, Brasil To Professor Antonio Plans. Abstract. We prove a representation theorem in terms of finite rank operators for operators belonging to £ . Some oo, oo, y
information on the tensor product of operators belonging to these ideals is also obtained.
INTRODUCTION. The n-th
approximation
number
a (T) of a bounded
linear
n
operator T€ J£(E,F) acting between the Banach spaces E and F, is defined as a (T) = inf { |T-T II: T € £(E,F), rank T < nl , n=l,2 n J n y " n" n (see [3], [5]) For 0 < y < oo the ideal
\£ ,
