2-summing operators and L (2)(2)-systems

Arch. Math. 71 (1998) 465 ± 471 0003-889X/98/060465-07 $ 2.90/0  Birkhäuser Verlag, Basel, 1998

Archiv der Mathematik

2-summing operators and L…2†-systems By FRANZISKA BAUR

Abstract. Given an infinite orthonormal system B in some L2 …m† we denote by P B the Banach ideal of B-summing operators. We show that P B coincides with the Banach ideal P 2 of 2-summing operators with equivalent norms if and only if B fails to be a uniformizable L…2†-system and with equal norms if and only if B fails to be a L…2†-system.

We use standard Banach space notation. In particular, for any Banach space X we denote by BX its closed unit ball, by X  its topological dual and by idX the identity operator on X. For the Banach ideals of bounded and 2-summing operators we use the standard notations ‰l; k
kŠ and ‰P 2 ; p2 Š, respectively. Given two Banach ideals ‰a; aŠ and ‰b; bŠ we write a  b if we have a…X; Y†  b…X; Y† for all pairs of Banach spaces X and Y. If a  b, then there exists a constant c > 0 such that b…T† % c
a…T† for all Banach spaces X, Y and all T 2 a…X; Y† (see [5] 6.1.6). If we can choose c ˆ 1 we write ‰a; aŠ  ‰b; bŠ. In the natural fashion, this leads to the notations a ˆ b and ‰a; aŠ ˆ ‰b; bŠ. Henceforth, we shall consider probability spaces …W; S; m† such that L2 …m† is infinite and B  L2 …m† will always denote an infinite orthonormal system. D e f i n i t i o n 1 . An operator T: X ! Y is said to belong to the class P B …X; Y† of B-summing operators if there exists a constant c > 0 such that for all finite sequences …bi †n1 in B and …xi †n1 in X we have: n 1=2 1=2  n „ P P  …1† k bi …w†Txi k2 dm…w† % c
sup jhx ; xi ij2 : W

iˆ1

x 2BX 

iˆ1

We write pB …T† for the smallest constant c satisfying …1†. Also, we abbreviate the . supremum on the right hand side of …1† by k…xi †niˆ1 kweak 2 The B-summing operators form a Banach ideal which will be denoted by ‰P B ; pB Š. Within this setting the classical Banach ideal ‰P 2 ; p2 Š of 2-summing operators can be described  n in many ways. If we choose for example the orthonormal system Bc :ˆ 22 cn : n 2 N  L2 ……0; 1†† where cn denotes the characteristic function of the interval …2ÿn ; 21ÿn †, then we have ‰P 2 ; p2 Š ˆ ‰P Bc ; pBc Š. Moreover, by a result due to M. Defant and M. Junge [3] such an equality holds whenever B is an orthonormal basis in L2 …m†. Mathematics Subject Classification (1991): Primary 47B10, 42A46; Secondary 47D50, 42C05.

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F. BAUR

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From this it is immediate that ‰P 2 ; p2 Š  ‰P B ; pB Š for every infinite orthonormal system B. In order to characterize those orthonormal systems B for which P B coincides with P 2 isomorphically and isometrically, respectively, we need the following notions. D e f i n i t i o n 2 . (i) An orthonormal system B in L2 …m† is said to be a L…2†-system if there exists a constant F ^ 1 such that …2†

kf k2 % F
kf k1

for all f 2 span B. The least constant F satisfying (2) is denoted by F…B; 2; 1† and is called the L…2; 1†-constant of B. (ii) An orthonormal system B  L2 …m† is called a uniformizable L…2†-system if for each " > 0 we can find a constant c ˆ c…B; "† > 0 such that for every f 2 span B there is a function g 2 L 1 …m† with the properties (a) kgk 1 % c
kf k2 (b) kg ÿ f k2 % "
kf k2 . Every uniformizable L…2†-system is a L…2†-system. As we will see soon the converse does not hold in our general setting. In the special setting of groups however, this is a longstanding open problem. Given an orthonormal system B in L2 …m† we put  WB;2 :ˆ j f j2 : f 2 span B ; k f k2 ˆ 1 : It was shown by Fournier in [4] that a subset L of the dual group G of a compact abelian group is a uniformizable L…2†-system if and only if WL;2 is uniformly integrable. This has the following straightforward generalization. Theorem 3. An orthonormal system B  L2 …m† is a uniformizable L…2†-system if and only if WB;2 is uniformly integrable. From [1] we know that P B coincides with P 2 if and only if WB;2 fails to be uniformly integrable. This leads to the following Theorem 4. An orthonormal system B  L2 …m† fails to be a uniformizable L…2†-system if and only if P B ˆ P 2. We are going to show that B fails to be a L…2†-system if and only if ‰P B ; pB Š ˆ ‰P 2 ; p2 Š. For this we introduce the following notation. Let " > 0 be given. We say that a subset W  L1 …m† has property 8 < for every d > 0 there exist A 2 S and w 2 W „ …U" † if : such that m…A† % d and jwj dm > " : A

Hence, by Theorem 3, an orthonormal system B  L… m† is a uniformizable L…2†-system if and only if WB;2 fails to have property …U" † for all " > 0. As we will see, B is a L…2†-system if and only if WB;2 fails …U" † for some " > 0.

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Lemma 5. Let 0 < " < 1 and B an orthonormal system in L2 …m†. If WB;2 has property …U" † and B0  B is finite, then WBnB0 ;2 has …U"0 † for all 0 < "0 < " . P r o o f. Fix 0 < "0 < ", B0 ˆ fb1 ; . . . ; bn g  B and assume that WBnB0 ;2 does not have „ …U"0 †. So there is a 0 < d0 < 1 such that jgj dm % "0 for all g 2 WBnB0 ;2 and all A 2 S A

satisfying m…A† % d0 . We choose 0 < "00 < 1 such that …"0 †1=2 ‡ …"00 †1=2
n1=2 % "1=2 : By uniform integrability of B20:ˆ fb21 ; . . . ; b2n g  L1 …m† we can find 0 < d < d0 such that „ jbi j2 dm % "00 for all i 2 f1; . . . ; ng and all A 2 S with m…A† % d. For any w 2 WB;2 there are A n P g 2 span B n B0 and ai 2 IK, i ˆ 1; . . . ; n such that if h :ˆ g ‡ ai bi, then jhj2 ˆ w. iˆ1
1=2 ÿP Moreover, kgk2 % khk2 ˆ 1 and jai j2 % khk2 ˆ 1. Now, given A 2 S with m…A† % d, i we have 1=2  „ 1=2 P „ 1=2 „ n w dm % jgj2 dm ‡ jai j jbi j2 dm A

iˆ1

A

A

% …"0 †1=2 ‡ …"00 †1=2
n1=2 % "1=2 : This contradicts our assumption.

h

This will now be applied to establish the following lemma which in turn will lead to a proof of the announced result. Lemma 6. Let B  L2 …m† be an orthonormal system. Then the following statements are equivalent. (i) B is not a L…2†-system. (ii) WB;2 has property …U" † for all 0 < " < 1. (iii) For all 0 < " < 1 and all n 2 IN we find orthonormal elements f1 ; . . . ; fn in span B and pairwise disjoint A1 ; . . . ; An 2 S such that „ jfi j2 dm > " for i ˆ 1; . . . ; n : Ai

P r o o f. …i† ) …ii† Let 0 < " < 1 and 0 < d < 1 be given. Choose 0 < l < d such that B is not a L…2†-system, we can find f 2 span B such that kf k1 % l and 1 ÿ l1=2 > ". Since  kf k2 ˆ 1. Set A :ˆ w 2 W : jf …w†j ^ lÿ1=2 . Then m…A†
lÿ1 %

„ A

jf j2 dm % 1

and so m…A† % l < d. Moreover, „ „ jf j2 dm % lÿ1=2
jf j dm % l1=2 : Ac

Consequently, „ A

Ac

jf j2 dm ^ 1 ÿ l1=2 > " :

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…ii† ) …i† Suppose that B is a L…2†-system with constant F ˆ F…B; 2; 1†. Choose 0 < d < 1 and 0 < " < 1 such that …1 ÿ "†1=2 ‡ d1=2 < Fÿ1 : By our hypothesis we find f 2 span B with kf k2 ˆ 1 and A 2 S such that m…A† % d and „ j f j2 dm > " : Now „ 1=2 A „ j f j dm % j f j2 dm
m…A†1=2 % d1=2 A

W

holds. For the complement Ac of A we get „ 1=2  1=2 „ „ j f j dm % j f j2 dm ˆ 1 ÿ j f j2 dm < …1 ÿ "†1=2 : Ac

Ac

A

This leads to a contradiction: k f k1 < …1 ÿ "†1=2 ‡ d1=2 < Fÿ1 % k f k1 : …ii† ) …iii† Let us fix 0 < " < "0 < 1, n 2 IN and positive numbers dj;k, k ˆ 2; . . . ; n, n P dj;k % "0 ÿ ". j ˆ 1; 2; . . . ; k ÿ 1 such that max 1 % j "0 : D1

„ Choose r2 > 0 such that jf1 j2 dm % d1;2 for all D 2 S with m…D† % r2 . Note that  D B1 :ˆ b 2 B : … f1 jb† ˆ j 0 is finite. By Lemma 5, WBnB1 ;2 has property …U"0 †. Accordingly, we can find a function f2 2 span B n B1 with k f2 k2 ˆ 1, and a D2 2 S such that „ m…D2 † % r2 and jf2 j2 dm > "0 : D2

„

2

Choose r3 > 0 such that j fi j dm % di;3, for all D 2 S with m…D† % r3 , i ˆ 1; 2. Note that  D B2 :ˆ b 2 B : … f2 jb† ˆ j 0 is a finite subset of B n B1. Another application of Lemma 5 yields a finite set B3  B n …B1 [ B2 †, a function f3 2 span B3 with k f3 k2 ˆ 1 and a D3 2 S such that „ m…D3 † % r3 and j f3 j2 dm > "0 : „

D3

We choose r4 > 0 such that j fi j2 dm % di;4 for all D 2 S with m…D† % r4 , i ˆ 1; 2; 3, and D continue. We end up with orthonormal elements f1 ; . . . ; fn in span B and sets D1 ; . . . ; Dn 2 S such that „ jfk j2 dm > "0 ; k ˆ 1; . . . ; n ; and

Dk

„ Dk

jfj j2 dm % dj;k ;

k ˆ 2; . . . ; n ; j ˆ 1; . . . ; k ÿ 1 :

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The sets Ak :ˆ Dk n [ Dj, 1 % k < n, and An :ˆ Dn are pairwise disjoint. Clearly jˆk‡1 „ jfn j2 dm ^ " and since also An

„ Ak

jfk j2 dm ^

„ Dk

jfk j2 dm ÿ

„

n P

jˆk‡1 Dj

jfk j2 dm > "0 ÿ

n P jˆk‡1

dk;j ^ "

for 1 % k < n, we are done. …iii† ) …ii† is obvious. h Now we are ready to prove the main result. Theorem 7. An orthonormal system B  L2 …m† fails to be a L…2†-system if and only if ‰P B ; pB Š ˆ ‰P 2 ; p2 Š. P r o o f. ª)º Let T 2 P B …X; Y†, n 2 N, x1 ; . . . ; xn 2 X and 0 < " < 1 be given. Choose " < "0 < 1 such that …"0 †1=2 ÿ n1=2 …1 ÿ "0 †1=2 ^ ". By Lemma 6 we find orthonormal elements „ j fi j2 dm > "0 for f1 ; . . . ; fn 2 span B and pairwise disjoint sets A1 ; . . . ; An 2 S such that Ai i ˆ 1; . . . ; n. We obtain !1=2 !1=2

n

2

n

2

„ P „ P

fi …w†Txi dm…w†

fi …w†c …w†Txi dm…w† ^ Ai



W

iˆ1

W

ÿ ^

iˆ1

!1=2

n

2

„ P

fi …w†c c …w†Txi dm…w† Ai

iˆ1

W 0

"


n P iˆ1

!1=2 2

kTxi k

ÿ

n P iˆ1

 kTxi k



ÿ ^ …"0 †1=2 ÿ n1=2 …1 ÿ "0 †1=2
 ^"


n P iˆ1

kTxi k2

1=2



n P iˆ1

„ Aci

1=2 jfi j2 dm

kTxi k2

1=2

:

N P Choose N 2 N, b1 ; . . . ; bN 2 B and ai;j 2 K, 1 % i % n, 1 % j % N such that fi ˆ ai;j bj , jˆ1 1 % i % n. Since f1 ; . . . ; fn are orthonormal elements we get !1=2

n

2 1=2  n

„ P P 2

kTxi k % "


fi …w†Txi dm…w† iˆ1

ˆ

W

iˆ1

W

jˆ1

!1=2

N  2  n

„ P

bj …w†T P ai;j xi dm…w†

iˆ1

 N

weak

P

n % pB …T† ai;j xi

iˆ1 jˆ1 % pB …T†
k…xi †niˆ1 kweak : 2

2

Hence, since 0 < " < 1 was arbitrary, p2 …T† % pB …T†. Since pB …T† % p2 …T† is always true we are done.

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ª(º By our hypothesis we have pB …idln1 † ˆ n1=2 for all n 2 N. We show that WB;2 has property …U" † for any 0 < " < 1. Accordingly, let us fix 0 < d < 1 and an auxiliary " < "0 < 1. Choose m 2 N such that m
d > 1 and then n 2 N such that m % n
…"0 ÿ "†. Our hypothesis yields the existence of finitely many x1 ; . . . ; xN 2 ln1 and b1 ; . . . ; bN 2 B such that

N

2

„ P weak

bj xj dm > n
"0 : k…xi †N k ˆ 1 and iˆ1 2

W

jˆ1

ln1

The latter is equivalent to „ max j fk j2 dm > n
"0 ; W 1%k%n

where fk :ˆ

N P

hxj ; ek ibj, k ˆ 1; . . . ; n. Moreover,  N 1=2 P 2 kfk k2 ˆ jhxj ; ek ij % 1 ; k ˆ 1; . . . ; n :

jˆ1

jˆ1

Next we take up an argument from [7] Lemma 31.3. Consider the sets n o Ak :ˆ w 2 W : k is the first l such that jfl …w†j ˆ max jft …w†j ; 1%t%n

n

k ˆ 1; . . . ; n. Note that they are measurable, pairwise disjoint, that W ˆ [ Ak and that n
"0
" , then Aj

0

n
" < jJj ‡ …n ÿ jJj†
" % jJj ‡ n
" and so jJj > n
…"0 ÿ "† ^ m. Since m is a probability measure there is a j 2 J „ jfj j2 dm > ", and so B fails to be a L…2†-system by such that m…Aj † % 1=m < d. But Aj Lemma 6. h We conclude with two remarks. R e m a r k 8 . (a) There exist L…2†-systems B for which P 2 ˆ P B holds only isomorphically but not isometrically. To see this, we consider a KasÆin decomposition of L2 …‰0; 1Š† (I am indebted to M. Junge for pointing this out to me): There exists an orthogonal decomposition L2 …‰0; 1Š† ˆ E1  E2 such that the L2 and L1 norms are equivalent on both E1 and E2 (E1 and E2 infinite dimensional) (cf. [6] p. 95). If we choose an orthonormal basis Bi in each Ei , then, B1 [ B2 being an orthonormal basis of L2 …‰0; 1Š†, the equality ‰P B1 ; pB Š ˆ ‰P 2 ; p2 Š holds. But this yields that either B1 or B2 fails to be uniformly integrable and so by Theorem 3 and Theorem 4 P B1 ˆ P 2 or P B2 ˆ P 2 , although B1 and B2 are L…2†-systems. (b) The Haar system is the system of functions h ˆ fcn : n 2 Ng, where c1 ˆ 1, and for 2k < n % 2k‡1 , k ˆ 0; 1; . . ., cn is defined as follows:

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2-summing operators and L…2†-systems

cn …x† :ˆ

8 > > > > > > > >
> > > > > > > : ÿ2k=2

for

n ÿ 2k ÿ 1 n ÿ 2k ; 2k 2k



n ÿ 2k ÿ 1 2…n ÿ 2k † ÿ 1 ; 2k 2k‡1   k 2…n ÿ 2 † ÿ 1 n ÿ 2k : x2 ; 2k 2k‡1



For all choices of a infinite subsystem B of the Haar system, the set WB;2 has the property …U1 †. Hence, by Lemma 6 and Lemma 7, P B coincides with P 2 with equal norms. The preceding results are contained in the authors thesis [2] written at the University of Zurich under the supervision of Prof. Dr. H. Jarchow. References [1] F. BAUR, Operator ideals, orthonormal systems and lacunary sets. Math. Nachr., to appear. [2] F. BAUR, Banach operator ideals generated by orthonormal systems. Dissertation, Universität Zürich (1997). [3] M. DEFANT and M. JUNGE, Unconditional orthonormal systems. Math. Nachr. 158, 233 ± 240 (1992). [4] J. J. F. FOURNIER, Uniformizable L(2) sets and uniform integrability. Colloq. Math. 51, 119 ± 128 (1987). [5] A. PIETSCH, Operator Ideals. Amsterdam 1980. [6] G. PISIER, The Volume of Convex Bodies and Banach Space Geometry. Cambridge 1989. [7] N. TOMCZAK-JAEGERMANN, Banach-Mazur Distances and Finite-Dimensional Operator Ideals. Pitman Monographs Surveys Pure Appl. Math. 38 (1989). Eingegangen am 16. 12. 1997 Anschrift der Autorin: Franziska Baur School of Mathematics University of New South Wales Sydney 2052 Australia