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Weak–Strong Continuity of Multilinear Mappings and the Pełczynski–Pitt Theorem
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
206, 532]546 Ž1997.
AY975253
Weak]Strong Continuity of Multilinear Mappings and the Pełczynski]Pitt Theorem ` Raymundo Alencar U IMECC-Unicamp, 13.081-970 Campinas, Sao ˆ Paulo, Brasil
and Klaus FloretU FB Mathematik der Uni¨ ersitat, ¨ 26111 Oldenburg, Germany Submitted by Richard M. Aron Received August 5, 1994
In generalizing a result of Pełczynski ` the sequential weak to norm continuity of N-linear mappings between certain Banach spaces Žincluding spaces of type and cotype. will be studied. In particular, it is shown that every N-linear continuous mapping from l p 1 = ??? = l p N into l q is compact if and only if Ý nis 1Ž1rpi . - 1rq. Q 1997 Academic Press
1. INTRODUCTION The result about mappings from l p 1 = ??? = l p N into l q was expected for a while; for polynomials it was settled by Pełczynski ` w19x in 1957. Pitt w26x had formulated his result in 1936 for bilinear mappings from l p = l q into K Žscalars ., but we do not know of any direct generalization of his arguments from 2 to N. Pełczynski ` gave in the above-mentioned paper a result about N-linear mappings from E N s^E =` ??? = E _ into F with a N
somehow involved proof, but nevertheless it is possible to modify his ideas to obtain the result for N-linear mappings from E1 = ??? = EN into F. We * The authors thank the Brazilian-German CNPqrGMD-agreement and FAEP-UNICAMP for support. 532 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
CONTINUITY OF MULTILINEAR MAPPINGS
533
shall, however, not give these Žstill a bit complicated. modifications and prove a slightly better result, using the generalized Rademacher functions Ždue to Aron and Globevnik w3x. as Aron et al. w4x did in their proof of Pełczynski’s result. ` For polynomials, Gonzalo and Jaramillo w17x have obtained results in the same spirit Žalso using N-Rademacher functions., but they used upper and lower estimates for sequences instead of Pełczynski’s ta-convergence Žsee ` 2.1 and 2.2 below.. We feel, however, that the ta-convergence is more natural in the context of weak]strong continuity of N-linear mappings than the upper and lower estimates, though the two concepts are closely related Žsee 2.2.. 2. THE ta-CONVERGENCE 2.1. Our notation will be standard; in particular, we denote by LŽ E1 , . . . , EN ; F . the space of continuous N-linear mappings from E1 = ??? = EN into F and by PN Ž E; F . the space of continuous N-homogeneous polynomials from E into F. Let E be a Žreal or complex. normed space and 0 F a - 1; a sequence Ž x n . in E is called ta-con¨ ergent to 0 if there exists a constant c G 0 such that
Ý
xn F c< B< a
ngB
for all finite subsets B ; N. It is straightforward to see that this is equivalent to
Ý
ln x n F c < B < a
ngB
for all finite subsets B ; N and l n g K with < l n < F 1 Žthe constant c may change.. A sequence Ž x n . in E is called ta-con¨ ergent to x g E if Ž x n y x . is ta-convergent to 0. 2.2. It is easy to see that the ta-convergence implies the s Ž E, EX .N convergence and that the Cesaro means Ny1 Ý ns 1 x n converge to x. The w notation of ta-convergence is due to Pełczynski 21x and it is closely related ` to the concept of upper and lower estimates of a sequence. A sequence Ž x n . in a Banach space E has an upper p-estimate Ž1 - p - `. if there is a c G 0 such that `
Ý an x n ns1
Fc
`
1rp
< an < p
žÝ / ns1
for all Ž a n . g l p
534
ALENCAR AND FLORET
Žthis is the same as Ž x n . being weakly pX-summable.; if the converse inequality holds for some constant c ) 0 we say that the sequence has a lower p-estimate. PROPOSITION.
Let Ž x n . be a sequence in a Banach space E.
If Ž x n . has an upper p-estimate then Ž x n . is t 1r p con¨ ergent to 0. If Ž x n . has a lower q-estimate then 5Ý ng B x n 5 G c < B < 1r q. Ž3. If Ž x n . is ta-con¨ ergent to 0 and 1 - p - 1ra then Ž x n . has an upper p-estimate. Ž4. If Ž x n . is an unconditional, semi-normalized Ž i.e., 0 - c1 F 5 x n 5 F c 2 - ` for some constants. basic sequence with 5Ý ng B x n 5 G c < B < a for all finite B ; N, then Ž x n . has lower q-estimates for all q ) 1ra . Ž1. Ž2.
Proof. Parts Ž1. and Ž2. are obvious, Ž3. is Lemma 3.4 of w15x and Ž4. is Lemma 1.4 of w14x. We shall not use this proposition, but found it necessary to state it in view of the results in w14x and w17x. 2.3. A normed space E has rank a g w0, 1w if the ta-convergence in E implies the convergence with respect to the norm. E has loose rank a gx0, 1x if it has rank r for all 0 F r - a . This concept is due to Pełczynski ` w21x as well. We say that E has property Pa if every s Ž E, EX .-null sequence admits a ta-convergent subsequence; it is clear that this property has to be checked only for seminormalized weak null sequences or even only for weak null sequences in the unit sphere. For a ) 0 it will be shown in 5.8 that this property is equivalent to the Banach]Saks type 1ra of E. Note that Schur spaces Žweak s norm convergence for sequences . have all Pa and loose rank 1. The following result was given, more or less, by Pełczynski ` in w21x. PROPOSITION. Ž1. E¨ ery finite dimensional space has loose rank 1. Ž2. The space l p Ž for 1 F p - `. has loose rank 1rp, but does not ha¨ e rank 1rp. Ž3. The space c 0 has no rank. E¨ ery Banach space not containing c 0 has rank 0. Ž4. l p has property P1r p for 1 - p - ` and c 0 has property P0 . Proof. Ž1. is immediate from the weak convergence. If Ž x n . is a seminormalized weak null sequence in l p or c 0 , a usual gliding hump argument gives that there is a subsequence Ž xs Ž n. . which is equivalent to the unit
535
CONTINUITY OF MULTILINEAR MAPPINGS
vector basis: xs Ž n. ;
Ý ngB
e n s < B < 1r p
Ý ngB
This easily implies Ž2., Ž4. and the first part of Ž3.. The second part of Ž3. is a direct consequence of a result of Bessaga]Pełczynski ` about unconditional series Žsee w12, p. 45x.. It is clear that a normed space where every seminormalized weakly null sequence Ž x n . has a subsequence Ž yn . with < B < a F c 5Ý ng B yn 5 for all finite B ; N has loose rank a . The identity map of a Banach space is absolutely Ž q, 1.-summing if there is a c ) 0 with 1rq
n
ž
Ý 5 xk 5 q ks1
/
F c sup
½
n
Ý lk x k
< lk < F 1
ks1
5
for all x 1 , . . . , x n g E. Now it is immediate that Ž5. A Banach space with absolutely Ž q, 1.-summing identity has loose rank 1rq. In particular: e¨ ery space with the Orlicz-property has loose rank 1r2 and e¨ ery space of Ž Rademacher . cotype q has loose rank 1rq. The latter follows e.g. from w10, p. 103 and p. 322x. the observation about cotype seems to be due to Castillo and Sanchez w8x and will be improved in Theorem 5.1.
3. THE MAIN THEOREM 3.1. If a s Ž a 1 , . . . , a N . g w0, 1w N is a multi-index, a sequence Ž x n . in E1 = ??? = EN is called ta-convergent if the sequence Ž x nŽ i .. ng N is N ta i-convergent in Ei for i s 1, . . . , N. Recall the convention < a < [ Ý is1 ai. MAIN THEOREM. Let E1 , . . . , EN and F be normed spaces, A : E1 = ??? = EN ª F an N-linear and continuous mapping, and a g w0, w N . If Ž x n . in E1 = ??? = EN is ta-con¨ ergent to x then Ž AŽ x n .. is t < a
536
ALENCAR AND FLORET
While the passage of Pełczynski’s proof to different spaces is not at all ` straightforward, the proof with Rademacher functions can be easily generalized; we give the details for the sake of completeness. Proof. Ža. We shall prove the result first for zero-sequences in complex spaces. The N-Rademacher functions of Aron and Globevenik w3x sn : w0, 1x ª l1 , . . . , l N 4 Žwhere l i are the Nth unit roots. are step functions with the following property: 1
H0 s
Ž t . . . . sn N Ž t . dt s
n1
½
n1 s ??? s n N otherwise.
1 0
Now take Ž x n . a ta-null sequence in E1 = ??? = EN and B ; N finite:
Ý AŽ x n . ngB
F
1
s
Ý
n 1 , . . . , n N gB
F
1
A
H0
žÝ
H0 s
n 1gB
F 5 A5
1
Ž t . . . . sn N Ž t . dt A Ž x n1Ž 1 . , . . . , x n N Ž N . .
n1
F
sn1Ž t . x n1Ž 1 . , . . . ,
N
H0 Ł Ý js1
n jgB
Ý
n N gB
sn N Ž t . x n N Ž N .
/
dt F
sn jŽ t . x n jŽ j . dt F 5 A 5 c1 . . . c N < B < < a < ,
by 2.1. Žb. For the passage from ta-null sequences to arbitrary ta-convergent sequences, we use the notation EJ [
Ł En , PJ Ž x . [ Ž x Ž j . . jgJ g EJ
ngJ
for x g EI
for f / J m I [ 1, 2, . . . , N 4 , and for y g E J and A : EI ª F we define A Jy : EI _ J ª F to be the mapping A with the jth components being fixed by y Ž j . for all j g J. With this notation we have AŽ x n q x . s AŽ x n . q AŽ x . q
Ý
f/JmI
A PJ J Ž x . Ž PI _ J x n .
for all x n , x g EI . If x n ª 0 Žta ., then part Ža. of the proof shows that AŽ x n q x . y AŽ x . is t < a <-convergent to 0. Žc. The real case is easily deduced from the complex one observing that every continuous N-linear mapping A : E1 = ??? = EN ª F has a ˆ where Eˆ continuous complex N-linear extension Aˆ : Eˆ1 = ??? = EˆN ª F, is a complexification of E.
CONTINUITY OF MULTILINEAR MAPPINGS
537
3.2. From the main theorem and using the definitions the following result is obvious: COROLLARY 1. Suppose Ei , F are normed spaces, Ei has property Pa i and F has rank r Ž or loose rank r .. If < a < F r Ž or < a < - r . then e¨ ery N-linear continuous mapping E1 = ??? = EN ª F is sequentially weak to norm continuous. This type of continuity has various consequences; e.g. such mappings map weakly sequentially compact Žs weakly compact. sets onto normcompact ones. This implies part Ž1. of COROLLARY 2. If, additionally, to the assumptions of corollary 1, all Ei and F are reflexi¨ e then
i¨ e.
Ž1.
All A g LŽ E1 , . . . , EN ; F . are compact.
Ž2.
˜p F X are reflexThe spaces LŽ E1 , . . . , EN ; F . and E1mp . . . mp EN m
&
Proof. To see Ž2. take x n g BE1 = ??? = BEN such that 5 AŽ x n .5 ª 5 A 5, choose a weakly convergent subsequence with limit x, then Žby corollary 1. 5 AŽ x .5 s 5 A 5. James’ sup-theorem Žsee e.g. w16x. gives that E [ E1 & ˜p F X is reflexive since EX s LŽ E1 , . . . , EN ; F .. mp . . . mp EN m Note the special case where F s K and < a < - 1. For polynomials one obtains the following COROLLARY 3. Let E be a Banach space with property Pa and F a Banach space with rank r Ž or loose rank r . such that Na F r Ž or Na - r ., then Ž1. E¨ ery continuous N-homogeneous polynomial E ª F is sequentially weak to norm continuous. If, additionally, E and F are reflexi¨ e, then Ž2. All P g PN Ž E; F . are compact. Ž3. PN Ž E; F . is reflexi¨ e. We note that the results given on spaces l p in section 4 imply that the conditions < a < - r in Corollary 1 and Na - r in Corollary 3 cannot be improved in general. 3.3. For F s K it follows that the completed N-fold symmetric ˜pN, s E of E is reflexive if E is reflexive and has projective tensor produce m the property Pa with Na - 1; moreover, by corollary 2 the completed ˜pN E is reflexive, which has m ˜pN, s E as a N-fold projective tensor product m N ˜p , s E implies subspace. In general it is not clear whether the reflexivity of m ˜pN E. the reflexivity of m
538
ALENCAR AND FLORET
This is true, however, at least if E is isomorphic to E N by the following PROPOSITION. Let E be a locally con¨ ex space, then mpN E is isomorphic to a complemented subspace of mpN, s E N. For N s 2 this result is due to Bonet and Peris w7x; the dual result that LŽ E, . . . , E; K. is isomorphic to a complemented subspace of PN Ž E N ; K. was observed by Defant and Maestre w11x. Proof. Let Ii : E ª E N and Pi : E N ª E be the ith canonical injection and projection. Define, with the usual Rademacher functions r j ,
s Ž y 1 , . . . , yN . [
1 N!
1
H0
N
r 1 Ž t . . . . rN Ž t . mN
Ý r i Ž t . yi
dt g msN E N
is1
Žpolarization formula. for y 1 , . . . , yN g E N and let sˆ : mN E N ª msN E N be its linearization. Now, if J : msN E N ª mN E N is the canonical injection, then the mapping J
msN E N ¨ mN E N
P1m ? ? ? mPN
mN E
6
sˆ
6
mN E I1m ? ? ? mIN mN E N
is 1rN! id. Since all these mappings are continuous with respect to the projective topologies, this gives the result. 3.4. The main theorem gives the sequential weak to norm continuity of multilinear mapping and polynomials in many cases. Therefore it is worthwhile to note the following observation which was made in w2x for polynomials. PROPOSITION. If the Ei are normed spaces, Ž X, t . is a topological space and the mapping f : E1 = ??? = EN ª X is sequentially weak to t-continuous, then f is weak to t-continuous on e¨ ery weakly compact subset K of the normed space E1 = ??? = EN . Proof. This follows immediately from the fact that the weak topology on K is angelic, hence subsets of K are weakly closed if they are weakly sequentially closed Žsee e.g. w16, p. 39 and 30x.. 3.5. An N-linear mapping A : E1 = ??? = EN ª F is approximable if it can be approximated Žwith respect to the norm of N-linear operators. by finite combinations of w 1 m ??? m w N m y where w i g EiX and y g F. PROPOSITION. Let Ei and F be reflexi¨ e spaces, Ei with the approximation property and property Pa , and F with rank r Ž or loose rank r .. If < a < F r Ž or < a < - r . then e¨ ery A g LŽ E1 , . . . , EN ; F . is approximable.
539
CONTINUITY OF MULTILINEAR MAPPINGS
Proof. Since LŽ E1 , . . . , EN ; F . s LŽ E1; LŽ E2 , . . . , EN ; F .. is reflexive by corollary 2, it follows from a result of Holub Žsee e.g. w10x, p. 196. that every A g LŽ E1; LŽ E2 , . . . , EN ; F .. is compact, hence approximable. Now induction easily gives the result. Note that we actually showed that all A g LŽ E1 , . . . , EN ; F . are approximable if all Ei have the approximation property and LŽ E1 , . . . , EN ; F . is reflexive.
` 4. THE PEŁCZYNSKI]PITT THEOREM 4.1. For the sequence spaces l p our results give the PROPOSITION.
Let 1 - pi - `, then the following are equi¨ alent:
Ža. - 1. Žb. E¨ ery N-linear continuous mapping l p = ??? = l p ª K is weakly 1 N sequentially continuous. N Ý is1 1rpi
Žc. Žd.
&
l p 1mp . . . mp l p N is reflexi¨ e. LŽ l p 1, . . . , l p N ; K. is reflexi¨ e.
Proof. Žc. and Žd. are clearly equivalent. Since l p has property P1r p the implications Ž a. ª Ž b . ª Ž c . were shown in 3.2. So it remains to show N that Žc. implies Ža.. Assume Ý is1 1rpi G 1, then the Holder inequality ¨ guarantees that the canonical multiplication operator F : l p 1 = ??? = l p N ª l 1 , F Ž Ž j n1 . n , . . . , Ž j nN . n . [ Ž j n1 ??? j nN . n &
ˆ : l p mp . . . mp l p ª l 1 is well defined and onto hence its & linearization F 1 N is onto as well. It follows that l p 1mp . . . mp l p N cannot be reflexive. X 4.2. Observing that LŽ l p 1, . . . , l p N ; l q . s LŽ l p 1, . . . , l p N , l q ; K. holds isometrically, one obtains the equivalence of Ža. and Žd. of the
THEOREM.
Let pi , q gx1, `w, then the following statements are equi¨ alent:
N Ža. Ý is1 1rpi - 1rq. Žb. E¨ ery A g LŽ l p , . . . , l p ; l q . is compact. 1 N Žc. E¨ ery A g LŽ l p , . . . , l p ; l q . is sequentially weak to norm contin1 N uous. Žd. LŽ l p , . . . , l p ; l q . is reflexi¨ e. 1 N
This generalizes Pitt’s theorem for linear operators l p ª l q to multilinear operators, one might call it the Pełczynski-Pitt theorem. `
540
ALENCAR AND FLORET
Proof. Corollary 1 in 3.2 Žand 2.3. gives that Ža. ª Žc. ª Žb.. That Žb. ª Ža. follows again by using the canonical multiplication operator N Ž l p 1 = ??? = l p N ª l q Žif Ý is1 1rpi . G 1rq . which is not compact. 4.3. For p1 s ??? s pN s p, the condition on the indices is N - p Žin 4.1. and Nq - p Žin the Pełczynski-Pitt theorem.. It is clear Žby what ` was said in 3.2 and by the fact that l p ( l pN . that one has equivalent conditions on polynomials as stated in corollary 3 of 3.2.
5. TYPE, COTYPE AND THE BANACH-SAKS PROPERTY 5.1. Up to now the main applications of the results in section 3 were about sequence spaces l p . Moreover there are many obvious applications if the Ei are subspaces of c 0 Žhence have P0 . and F has some rank. Spaces of ŽRademacher. type and cotype Žsee w10x or w25x for the definition. can also be used, since the following result holds. THEOREM.
Let 1 - p F 2 F q - `.
Ž1. E¨ ery Banach space of type p has property P1r p . Ž2. E¨ ery Banach space E of cotype q has loose rank 1rq. E¨ en more: e¨ ery seminormalized weakly null sequence in E has a subsequence Ž x n . satisfying c < B < 1r q F
Ý
xn
ngB
for some c ) 0 and all finite B ; N. The proof will use the technique of spreading models which was already successfully applied to this kind of problems by Rakov w27, 28x, Farmer w14x, Farmer and Johnson w15x, and Gonzalo and Jaramillo w17x. In particular, Rakov showed in w28x the result Ž1. for the formally weaker Banach-Saks type p Žbut see 5.8 below. and Gonzalo and Jaramillo w17x have the related theorem for upper and lower estimates. 5.2. Spreading models allow to choose subsequences which are nearly unconditionally basic sequences. LEMMA 1. Let Ž e n . be a seminormalized unconditional basic sequence in a Banach space F. If F has type p then Ž e n . is t 1r p-con¨ ergent to 0. If F has cotype q then there is a constant c such that < B < 1r q F c 5Ý ng B e n 5 for all finite B ; N. Ž1. Ž2.
541
CONTINUITY OF MULTILINEAR MAPPINGS
Proof. There is a constant Žsee e.g. w10, p. 358x. such that N
1
N
Ý < ln < en
c
N
Ý ln en
F
ns1
Ý < ln < en
Fc
ns1
,
ns1
in particular, 1
Ý
c
en F
Ý rn Ž t . e n
ngB
Fc
ngB
Ý
en
ngB
for the Rademacher functions rn : w0, 1x ª y1, 14 . Therefore the definitions of type and cotype give the result. 5.3. If Ž x n . is a seminormalized s Ž E, EX .-null sequence then it has a subsequence Ž yn . with a spreading model Ž F, Ž e n .., i.e. a Banach space F with an unconditional seminormalized basis Ž e n . such that: Ža. For all « ) 0, all N g N and all l1 , . . . , l N g K there is an m g N such that N
N
Ý l k yn ks1
y
k
Ý lk en
F«
k
ks1
E
for all m F n1 - ??? - n N
F
Žb. For all n1 - n 2 - ??? - n N and scalars l k the equality N
N
Ý lk en
s
k
ks1
Ý lk ek ks1
F
. F
holds. For a proof of this, see w6x. LEMMA 2.
Let 0 - a - 1.
Ž1. If Ž e n . ta-con¨ erges to 0 then Ž yn . has a subsequence Ž z n . which ta-con¨ erges to 0. Ž2. If < B < a F c 5Ý ng B e n 5 F for some c ) 0 and all finite B ; N, then Ž yn . has a subsequence which satisfies the same type of lower estimate. It is easy to see that the converse statements in Ž1. and Ž2. hold as well. Proof. Ž1. By property Ža. of the spreading model for « s l1 s ??? s l N s 1 there is a strictly monotone function b : N ª N such that for all B ; N with < B < s N and min B G b Ž N .
Ž ).
Ý ngB 1r a x.
Define t Ž n. [ b Žw n and z n [ yt Ž n. .
F
yn E
Ý ngB
q 1 F c< B< a
en F
Žwhere w x x denotes the integral part of x g w1, `..
542
ALENCAR AND FLORET
For a given B ; N with < B < s N take m [ w N a x q 1, B0 [ w1, mwlB, B1 [ B_ B0 and choose D ; w m, `wlN_ B1 such that < D < s < B0 < F m y 1 F N a. Then
Ý
F
yt Ž n.
ngB
q
yt Ž n.
Ý ngB 0
E
q
yt Ž n.
Ý ngDjB1
E
Ý
yt Ž n.
ngD
E
E
F Ž < D < q < B0 < . sup 5 yn 5 E q cN a F ˜ cN a by Ž). since min t Ž D j B1 . G t Ž m. G b Ž N .. Ž2. The same b as above gives that
Ý
F
en
ngB
F
q1
yn
Ý ngB
E
for all B ; N with < B < s N and min B G b Ž N .. This time, the index for the subsequence will be defined by t Ž n. [ b w n2r a x. Proceeding as before but with m [ w N a r2 x q 1 hence t Ž m. G b Ž N ., we obtain c< B< a F
Ý
F
Ý
s
en
ngB
F
Ý
q
yt Ž n.
ngB
F
q
yt Ž n. E
Ý Ý
E
q1
yt Ž n.
ngB 0
q1
yt Ž n.
ngDjB1
E
q Ž < D < q < B0 < . sup 5 yn 5 E q 1.
yt Ž n.
ngB
Ý ngD
E
F
en
Ý ng t Ž DjB1 .
F
E
Since < D < s < B0 < F m y 1 F N a r2 one obtains for some constant c1
Ý
G c < B < a y c1 < B < a r2 .
yt Ž n.
ngB
E
Passing again to subsequence, one may assume that Ž yt Ž n. . is a basic Žstill seminormalized. sequence, hence there is a g ) 0 with g F 5Ý ng B yt Ž n. 5 E for all finite f / B ; N. It follows that
Ý ngB
G max g , c < B < a y c1 < B < a r2 4
yt Ž n. E
which easily implies a lower estimate of the form c 2 < B < a.
CONTINUITY OF MULTILINEAR MAPPINGS
543
5.4. Now everything is ready for the Proof of Theorem 5.1: Ž1. Let Ž x n . be a s Ž E, EX .-null seminormalized sequence with spreading model Ž F, Ž e n ... Since F is finitely represented in E Žsee w6x. it has also type p and lemmas 1 Ž1. and 2 Ž1. give the result. Ž2. Take a seminormalized ta-null sequence Ž x n . with a spreading model Ž F, Ž e n ... Since F has cotype q one obtains from Lemmas 1 Ž2. and 2 Ž2. that there is a subsequence Ž xt Ž n. . with c1 < B < 1r q F 5Ý ng B xt Ž n. 5 E . Adapting Lemma 1 Ž2., it is obvious that}as in 2.3 Ž5. }the statement Ž2. of the theorem holds also if id E is absolutely Ž q, 1.-summing. 5.5. The main theorem, for example, implies that for 1 - p F 2 F q - `, all A g LŽ L q w0, 1x, l 3 ; l 9r8 . and all a g LŽ l 5 , l 5 ; L p w0, 1x. are sequentially weak to norm continuous and compact and these spaces of operators are reflexive, since L q has type 2 and L p has cotype 2. Using the fact that l 2 ; L r w0, 1x is complemented for all 1 - r - ` the results in sections 3 and 4 imply equivalences of the kind L Ž L q w 0, 1 x , l 3 ; l s . is reflexive if and only if 1 - s - 6r5 L Ž l 5 , l s ; L p w 0, 1 x . is reflexive if and only if 10r3 - s - `, and in these cases all the operators are compact. 5.6. A Banach space of type p ) 1 has some cotype Žsee w25x, p. 39., hence one obtains the COROLLARY.
E¨ ery Banach space of type p ) 1 has positi¨ e rank.
It follows that every N-linear continuous mapping on c 0 with values in a Banach space of some type p ) 1 or of some cotype is sequentially weak to norm continuous and throws weakly compact subsets of c 0N onto norm compact sets. 5.7. Theorem 5.1 holds Žwith a weaker estimate in Ž2.. also for weak type and weak cotype Žsee w19, 20x for definitions. }this was suggested to us by M. Junge. PROPOSITION. Ž1. Ž2.
Let 1 - p - 2 F q - `.
E¨ ery Banach space of weak type p has P1r p . E¨ ery Banach space of weak cotype q has loose rank 1rq.
In particular, every Banach space of weak type 2 has Pa for all a gx1r2, 1w. Proof. It follows from w19, Theorem 3.4x Žand the fact that Rademacher-averages are smaller than Gauss-averages . that in spaces F with weak
544
ALENCAR AND FLORET
type p - 2 the inequality 1r2
2
n
žH Ý
rk Ž t . ek
ks1
dt F
/
F cn1r p
holds for all e k g F with 5 e k 5 s 1. Now one can proceed as in the proof of the theorem. The statement Ž2. follows directly from the theorem and the fact that weak cotype q implies cotype q q « for all « ) 0 Žthis follows from the results in w20x and w29, Theorem 12.2x.. 5.8. The same techniques of spreading models can be used to simplify the definition of the property Pa . In view of 2.2 Ž3. the following result improves w27, Theorem 3x, and w9, Theorem 2.3x. PROPOSITION. Let 0 - a - 1 and E be a normed space. Then E has property Pa if and only if E is of Banach-Saks type 1ra , i.e.: e¨ ery seminormalized s Ž E, EX .-null sequence admits a subsequence Ž x n . such that N
F cN a
xn
Ý ns1
E
for some c G 0 and all N g N. Proof. Assume that the condition is satisfied, take a seminormalized s Ž E, EX .-null sequence Ž x n . with spreading model Ž F, Ž e n .., choose a strictly monotone function b : N ª N such that y
xn
Ý ngB
for all B ; b Ž n . , b Ž n . q 1, . . . 4
F1
en
Ý ngB
E
F
with < B < s N and a subsequence Ž xt Ž n. . with t Ž n. G b Žw n1r a x. and N 5Ý ns1 xt Ž n. 5 E F cN a. Then, as in the proof of Ž2. in Lemma 2 with B s 1, . . . , N 4 , one obtains N
N
F
Ý en ns1
Ý ns1
F
q Ž < D < q < B0 < . sup 5 x n 5 E q 1 F ˜ cN a .
xt Ž n. E
Now the calculation in the proof of Ž1. in Lemma 2 and property Žb. of the spreading model give that
Ý ngB
F Ž < B0 < q < D < . sup 5 x n 5 E q
xt Ž n.
Ý ngDjB1
E
F c1 N a q
Ý
s c1 N a q
F
N
q 1 F Ž c1 q ˜ c q 1. N a .
Ý eN ns1
E
q1
en
ngDjB1
xt Ž n.
F
CONTINUITY OF MULTILINEAR MAPPINGS
545
5.9. It follows from w18x that for 0 - a - 1 an Orlicz sequence space h M not containing l 1 has property Pa if and only if the Orlicz function M satisfies: sup
½
M Ž st . M Ž s . t 1r a
: s, t g x 0, 1 x - `
5
Other conditions for Orlicz sequence spaces having property Pa appeared in w28x.
REFERENCES 1. R. Alencar and K. Floret, Weak continuity of multilinear mappings on Tsirelson’s space, Sem. Bras. Anal. ´ 40 Ž1994., 509]517. 2. R. M. Aron, C. Herves, and M. Valdivia, Weakly continuous mappings on Banach spaces, J. Funct. Anal. 52 Ž1983., 189]204. 3. R. M. Aron and J. Globevnik, Analytic functions on c 0 , Re¨ . Mat. Ž Madrid. 2 Ž1989., 27]34. 4. R. M. Aron, M. Lacruz, R. A. Ryan and A. M. Tonge, The generalized Rademacher functions, Note di Mat. 12 Ž1992., 15]25. 5. S. Banach, ‘‘Theorie des operations lineaires,’’ Chelsea Publishing Company, New York, ´ ´ 1978. 6. B. Beauzamy and T. J. Lapreste, ´ ‘‘Modeles ` ´etales ´ des espaces de Banach,’’ Hermann, Paris, 1984. 7. J. Bonet and A. Peris, On the injective tensor product of quasinormable spaces, Results in Math. 20 Ž1991., 431]443. 8. J. Castillo and F. Sanchez, Remarks on some basic properties of Tsirelson’s space, Note di Mat. 13 Ž1993., 117]122. 9. J. Castillo and F. Sanchez, Weakly p-compact, p-Banach-Saks, and super-reflexive Banach spaces, J. Math. Anal. Appl. 185 Ž1994., 256]261. 10. A. Defant and K. Floret, ‘‘Tensor Norms and Operator Ideals,’’ Math. Studies, Vol. 176, North Holland, Amsterdam, 1993. 11. A. Defant and M. Maestre, Property ŽBB. and holomorphic functions on Frechet]Montel ´ spaces, Math. Proc. Cambridge Philos. Soc. 115 Ž1993., 305]313. 12. J. Diestel, ‘‘Sequences and Series in Banach Spaces,’’ Graduate Texts in Math. Vol. 92, Springer-Verlag, New YorkrBerlin, 1984. 13. S. Dineen, ‘‘Complex Analysis in Locally Convex Spaces,’’ Math. Studies, Vol. 57, North Holland, Amsterdam, 1981. 14. J. Farmer, Polynomial reflexivity in Banach spaces, Israel J. Math. 87 Ž1994., 257]273. 15. J. Farmer and W. B. Johnson, Polynomial Schur and polynomial Dunford-Pettis properties, Contemp. Math. 144 Ž1993., 95]105. 16. K. Floret, ‘‘Weakly Compact Sets,’’ Lecture Notes in Math., Vol. 801, Springer-Verlag, New YorkrBerlin, 1980. 17. R. Gonzalo and J. Jaramillo, Compact polynomials between Banach spaces, Extracta Math. 8 Ž1993., 42]48. 18. H. Knaust, Orlicz sequence spaces of Banach-Saks type, Arch. Math. 59 Ž1992., 562]565. 19. V. Mascioni, On weak cotype and weak type in Banach spaces, Note di Mat. 8 Ž1988., 67]110.
546
ALENCAR AND FLORET
20. V. Mascioni and U. Matter, Weakly Ž q, p .-summing operators and weak cotype properties of Banach spaces, Proc. Roy. Irish Acad. 88A Ž1988., 169]177. 21. A. Pełczynski, A property of multilinear operations, Studia Math. 16 Ž1957., 173]182. ` 22. A. Pełczynski, ` On weakly compact polynomial operators on B-spaces with Dunford-Pettis property, Bull. Acad. Pol. Sci. 11 Ž1963., 371]378. 23. A. Pełczynski, A theorem of Dunford-Pettis type for polynomial operators, Bull. Acad. ` Pol. Sci. 11 Ž1963., 379]386. 24. A. Pełczynski ` and Z. Semadeni, Spaces of continuous functions ŽIII. ŽSpaces C Ž V . for V without perfect subsets ., Studia Math. 18 Ž1959., 211]222. 25. G. Pisier, ‘‘Factorization of Linear Operators and Geometry of Banach Spaces,’’ CBMS Regional Conference Series Vol. 60, Amer. Math. Soc., Providence, RI, 1986. 26. H. R. Pitt, A note on bilinear forms, J. London Math. Soc. 11 Ž1936., 174]180. 27. S. A. Rakov, Banach-Saks property of a Banach space, Math. Notes 26 Ž1980., 909]916 wtransl. from Mat. Zametkix. 28. S. A. Rakov, Banach-Saks exponent of certain Banach spaces of sequences, Math. Notes 32 Ž1982., 791]797 wtransl. from Mat. Zametkix. 29. N. Tomczak-Jaegermann, ‘‘Banach]Mazur Distances and Finite-Dimensional Operator Ideals,’’ Monographs and Sur¨ eys in Pure and Applied Math., Vol. 38, Pitman, London, 1989.
206, 532]546 Ž1997.
AY975253
Weak]Strong Continuity of Multilinear Mappings and the Pełczynski]Pitt Theorem ` Raymundo Alencar U IMECC-Unicamp, 13.081-970 Campinas, Sao ˆ Paulo, Brasil
and Klaus FloretU FB Mathematik der Uni¨ ersitat, ¨ 26111 Oldenburg, Germany Submitted by Richard M. Aron Received August 5, 1994
In generalizing a result of Pełczynski ` the sequential weak to norm continuity of N-linear mappings between certain Banach spaces Žincluding spaces of type and cotype. will be studied. In particular, it is shown that every N-linear continuous mapping from l p 1 = ??? = l p N into l q is compact if and only if Ý nis 1Ž1rpi . - 1rq. Q 1997 Academic Press
1. INTRODUCTION The result about mappings from l p 1 = ??? = l p N into l q was expected for a while; for polynomials it was settled by Pełczynski ` w19x in 1957. Pitt w26x had formulated his result in 1936 for bilinear mappings from l p = l q into K Žscalars ., but we do not know of any direct generalization of his arguments from 2 to N. Pełczynski ` gave in the above-mentioned paper a result about N-linear mappings from E N s^E =` ??? = E _ into F with a N
somehow involved proof, but nevertheless it is possible to modify his ideas to obtain the result for N-linear mappings from E1 = ??? = EN into F. We * The authors thank the Brazilian-German CNPqrGMD-agreement and FAEP-UNICAMP for support. 532 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
CONTINUITY OF MULTILINEAR MAPPINGS
533
shall, however, not give these Žstill a bit complicated. modifications and prove a slightly better result, using the generalized Rademacher functions Ždue to Aron and Globevnik w3x. as Aron et al. w4x did in their proof of Pełczynski’s result. ` For polynomials, Gonzalo and Jaramillo w17x have obtained results in the same spirit Žalso using N-Rademacher functions., but they used upper and lower estimates for sequences instead of Pełczynski’s ta-convergence Žsee ` 2.1 and 2.2 below.. We feel, however, that the ta-convergence is more natural in the context of weak]strong continuity of N-linear mappings than the upper and lower estimates, though the two concepts are closely related Žsee 2.2.. 2. THE ta-CONVERGENCE 2.1. Our notation will be standard; in particular, we denote by LŽ E1 , . . . , EN ; F . the space of continuous N-linear mappings from E1 = ??? = EN into F and by PN Ž E; F . the space of continuous N-homogeneous polynomials from E into F. Let E be a Žreal or complex. normed space and 0 F a - 1; a sequence Ž x n . in E is called ta-con¨ ergent to 0 if there exists a constant c G 0 such that
Ý
xn F c< B< a
ngB
for all finite subsets B ; N. It is straightforward to see that this is equivalent to
Ý
ln x n F c < B < a
ngB
for all finite subsets B ; N and l n g K with < l n < F 1 Žthe constant c may change.. A sequence Ž x n . in E is called ta-con¨ ergent to x g E if Ž x n y x . is ta-convergent to 0. 2.2. It is easy to see that the ta-convergence implies the s Ž E, EX .N convergence and that the Cesaro means Ny1 Ý ns 1 x n converge to x. The w notation of ta-convergence is due to Pełczynski 21x and it is closely related ` to the concept of upper and lower estimates of a sequence. A sequence Ž x n . in a Banach space E has an upper p-estimate Ž1 - p - `. if there is a c G 0 such that `
Ý an x n ns1
Fc
`
1rp
< an < p
žÝ / ns1
for all Ž a n . g l p
534
ALENCAR AND FLORET
Žthis is the same as Ž x n . being weakly pX-summable.; if the converse inequality holds for some constant c ) 0 we say that the sequence has a lower p-estimate. PROPOSITION.
Let Ž x n . be a sequence in a Banach space E.
If Ž x n . has an upper p-estimate then Ž x n . is t 1r p con¨ ergent to 0. If Ž x n . has a lower q-estimate then 5Ý ng B x n 5 G c < B < 1r q. Ž3. If Ž x n . is ta-con¨ ergent to 0 and 1 - p - 1ra then Ž x n . has an upper p-estimate. Ž4. If Ž x n . is an unconditional, semi-normalized Ž i.e., 0 - c1 F 5 x n 5 F c 2 - ` for some constants. basic sequence with 5Ý ng B x n 5 G c < B < a for all finite B ; N, then Ž x n . has lower q-estimates for all q ) 1ra . Ž1. Ž2.
Proof. Parts Ž1. and Ž2. are obvious, Ž3. is Lemma 3.4 of w15x and Ž4. is Lemma 1.4 of w14x. We shall not use this proposition, but found it necessary to state it in view of the results in w14x and w17x. 2.3. A normed space E has rank a g w0, 1w if the ta-convergence in E implies the convergence with respect to the norm. E has loose rank a gx0, 1x if it has rank r for all 0 F r - a . This concept is due to Pełczynski ` w21x as well. We say that E has property Pa if every s Ž E, EX .-null sequence admits a ta-convergent subsequence; it is clear that this property has to be checked only for seminormalized weak null sequences or even only for weak null sequences in the unit sphere. For a ) 0 it will be shown in 5.8 that this property is equivalent to the Banach]Saks type 1ra of E. Note that Schur spaces Žweak s norm convergence for sequences . have all Pa and loose rank 1. The following result was given, more or less, by Pełczynski ` in w21x. PROPOSITION. Ž1. E¨ ery finite dimensional space has loose rank 1. Ž2. The space l p Ž for 1 F p - `. has loose rank 1rp, but does not ha¨ e rank 1rp. Ž3. The space c 0 has no rank. E¨ ery Banach space not containing c 0 has rank 0. Ž4. l p has property P1r p for 1 - p - ` and c 0 has property P0 . Proof. Ž1. is immediate from the weak convergence. If Ž x n . is a seminormalized weak null sequence in l p or c 0 , a usual gliding hump argument gives that there is a subsequence Ž xs Ž n. . which is equivalent to the unit
535
CONTINUITY OF MULTILINEAR MAPPINGS
vector basis: xs Ž n. ;
Ý ngB
e n s < B < 1r p
Ý ngB
This easily implies Ž2., Ž4. and the first part of Ž3.. The second part of Ž3. is a direct consequence of a result of Bessaga]Pełczynski ` about unconditional series Žsee w12, p. 45x.. It is clear that a normed space where every seminormalized weakly null sequence Ž x n . has a subsequence Ž yn . with < B < a F c 5Ý ng B yn 5 for all finite B ; N has loose rank a . The identity map of a Banach space is absolutely Ž q, 1.-summing if there is a c ) 0 with 1rq
n
ž
Ý 5 xk 5 q ks1
/
F c sup
½
n
Ý lk x k
< lk < F 1
ks1
5
for all x 1 , . . . , x n g E. Now it is immediate that Ž5. A Banach space with absolutely Ž q, 1.-summing identity has loose rank 1rq. In particular: e¨ ery space with the Orlicz-property has loose rank 1r2 and e¨ ery space of Ž Rademacher . cotype q has loose rank 1rq. The latter follows e.g. from w10, p. 103 and p. 322x. the observation about cotype seems to be due to Castillo and Sanchez w8x and will be improved in Theorem 5.1.
3. THE MAIN THEOREM 3.1. If a s Ž a 1 , . . . , a N . g w0, 1w N is a multi-index, a sequence Ž x n . in E1 = ??? = EN is called ta-convergent if the sequence Ž x nŽ i .. ng N is N ta i-convergent in Ei for i s 1, . . . , N. Recall the convention < a < [ Ý is1 ai. MAIN THEOREM. Let E1 , . . . , EN and F be normed spaces, A : E1 = ??? = EN ª F an N-linear and continuous mapping, and a g w0, w N . If Ž x n . in E1 = ??? = EN is ta-con¨ ergent to x then Ž AŽ x n .. is t < a
536
ALENCAR AND FLORET
While the passage of Pełczynski’s proof to different spaces is not at all ` straightforward, the proof with Rademacher functions can be easily generalized; we give the details for the sake of completeness. Proof. Ža. We shall prove the result first for zero-sequences in complex spaces. The N-Rademacher functions of Aron and Globevenik w3x sn : w0, 1x ª l1 , . . . , l N 4 Žwhere l i are the Nth unit roots. are step functions with the following property: 1
H0 s
Ž t . . . . sn N Ž t . dt s
n1
½
n1 s ??? s n N otherwise.
1 0
Now take Ž x n . a ta-null sequence in E1 = ??? = EN and B ; N finite:
Ý AŽ x n . ngB
F
1
s
Ý
n 1 , . . . , n N gB
F
1
A
H0
žÝ
H0 s
n 1gB
F 5 A5
1
Ž t . . . . sn N Ž t . dt A Ž x n1Ž 1 . , . . . , x n N Ž N . .
n1
F
sn1Ž t . x n1Ž 1 . , . . . ,
N
H0 Ł Ý js1
n jgB
Ý
n N gB
sn N Ž t . x n N Ž N .
/
dt F
sn jŽ t . x n jŽ j . dt F 5 A 5 c1 . . . c N < B < < a < ,
by 2.1. Žb. For the passage from ta-null sequences to arbitrary ta-convergent sequences, we use the notation EJ [
Ł En , PJ Ž x . [ Ž x Ž j . . jgJ g EJ
ngJ
for x g EI
for f / J m I [ 1, 2, . . . , N 4 , and for y g E J and A : EI ª F we define A Jy : EI _ J ª F to be the mapping A with the jth components being fixed by y Ž j . for all j g J. With this notation we have AŽ x n q x . s AŽ x n . q AŽ x . q
Ý
f/JmI
A PJ J Ž x . Ž PI _ J x n .
for all x n , x g EI . If x n ª 0 Žta ., then part Ža. of the proof shows that AŽ x n q x . y AŽ x . is t < a <-convergent to 0. Žc. The real case is easily deduced from the complex one observing that every continuous N-linear mapping A : E1 = ??? = EN ª F has a ˆ where Eˆ continuous complex N-linear extension Aˆ : Eˆ1 = ??? = EˆN ª F, is a complexification of E.
CONTINUITY OF MULTILINEAR MAPPINGS
537
3.2. From the main theorem and using the definitions the following result is obvious: COROLLARY 1. Suppose Ei , F are normed spaces, Ei has property Pa i and F has rank r Ž or loose rank r .. If < a < F r Ž or < a < - r . then e¨ ery N-linear continuous mapping E1 = ??? = EN ª F is sequentially weak to norm continuous. This type of continuity has various consequences; e.g. such mappings map weakly sequentially compact Žs weakly compact. sets onto normcompact ones. This implies part Ž1. of COROLLARY 2. If, additionally, to the assumptions of corollary 1, all Ei and F are reflexi¨ e then
i¨ e.
Ž1.
All A g LŽ E1 , . . . , EN ; F . are compact.
Ž2.
˜p F X are reflexThe spaces LŽ E1 , . . . , EN ; F . and E1mp . . . mp EN m
&
Proof. To see Ž2. take x n g BE1 = ??? = BEN such that 5 AŽ x n .5 ª 5 A 5, choose a weakly convergent subsequence with limit x, then Žby corollary 1. 5 AŽ x .5 s 5 A 5. James’ sup-theorem Žsee e.g. w16x. gives that E [ E1 & ˜p F X is reflexive since EX s LŽ E1 , . . . , EN ; F .. mp . . . mp EN m Note the special case where F s K and < a < - 1. For polynomials one obtains the following COROLLARY 3. Let E be a Banach space with property Pa and F a Banach space with rank r Ž or loose rank r . such that Na F r Ž or Na - r ., then Ž1. E¨ ery continuous N-homogeneous polynomial E ª F is sequentially weak to norm continuous. If, additionally, E and F are reflexi¨ e, then Ž2. All P g PN Ž E; F . are compact. Ž3. PN Ž E; F . is reflexi¨ e. We note that the results given on spaces l p in section 4 imply that the conditions < a < - r in Corollary 1 and Na - r in Corollary 3 cannot be improved in general. 3.3. For F s K it follows that the completed N-fold symmetric ˜pN, s E of E is reflexive if E is reflexive and has projective tensor produce m the property Pa with Na - 1; moreover, by corollary 2 the completed ˜pN E is reflexive, which has m ˜pN, s E as a N-fold projective tensor product m N ˜p , s E implies subspace. In general it is not clear whether the reflexivity of m ˜pN E. the reflexivity of m
538
ALENCAR AND FLORET
This is true, however, at least if E is isomorphic to E N by the following PROPOSITION. Let E be a locally con¨ ex space, then mpN E is isomorphic to a complemented subspace of mpN, s E N. For N s 2 this result is due to Bonet and Peris w7x; the dual result that LŽ E, . . . , E; K. is isomorphic to a complemented subspace of PN Ž E N ; K. was observed by Defant and Maestre w11x. Proof. Let Ii : E ª E N and Pi : E N ª E be the ith canonical injection and projection. Define, with the usual Rademacher functions r j ,
s Ž y 1 , . . . , yN . [
1 N!
1
H0
N
r 1 Ž t . . . . rN Ž t . mN
Ý r i Ž t . yi
dt g msN E N
is1
Žpolarization formula. for y 1 , . . . , yN g E N and let sˆ : mN E N ª msN E N be its linearization. Now, if J : msN E N ª mN E N is the canonical injection, then the mapping J
msN E N ¨ mN E N
P1m ? ? ? mPN
mN E
6
sˆ
6
mN E I1m ? ? ? mIN mN E N
is 1rN! id. Since all these mappings are continuous with respect to the projective topologies, this gives the result. 3.4. The main theorem gives the sequential weak to norm continuity of multilinear mapping and polynomials in many cases. Therefore it is worthwhile to note the following observation which was made in w2x for polynomials. PROPOSITION. If the Ei are normed spaces, Ž X, t . is a topological space and the mapping f : E1 = ??? = EN ª X is sequentially weak to t-continuous, then f is weak to t-continuous on e¨ ery weakly compact subset K of the normed space E1 = ??? = EN . Proof. This follows immediately from the fact that the weak topology on K is angelic, hence subsets of K are weakly closed if they are weakly sequentially closed Žsee e.g. w16, p. 39 and 30x.. 3.5. An N-linear mapping A : E1 = ??? = EN ª F is approximable if it can be approximated Žwith respect to the norm of N-linear operators. by finite combinations of w 1 m ??? m w N m y where w i g EiX and y g F. PROPOSITION. Let Ei and F be reflexi¨ e spaces, Ei with the approximation property and property Pa , and F with rank r Ž or loose rank r .. If < a < F r Ž or < a < - r . then e¨ ery A g LŽ E1 , . . . , EN ; F . is approximable.
539
CONTINUITY OF MULTILINEAR MAPPINGS
Proof. Since LŽ E1 , . . . , EN ; F . s LŽ E1; LŽ E2 , . . . , EN ; F .. is reflexive by corollary 2, it follows from a result of Holub Žsee e.g. w10x, p. 196. that every A g LŽ E1; LŽ E2 , . . . , EN ; F .. is compact, hence approximable. Now induction easily gives the result. Note that we actually showed that all A g LŽ E1 , . . . , EN ; F . are approximable if all Ei have the approximation property and LŽ E1 , . . . , EN ; F . is reflexive.
` 4. THE PEŁCZYNSKI]PITT THEOREM 4.1. For the sequence spaces l p our results give the PROPOSITION.
Let 1 - pi - `, then the following are equi¨ alent:
Ža. - 1. Žb. E¨ ery N-linear continuous mapping l p = ??? = l p ª K is weakly 1 N sequentially continuous. N Ý is1 1rpi
Žc. Žd.
&
l p 1mp . . . mp l p N is reflexi¨ e. LŽ l p 1, . . . , l p N ; K. is reflexi¨ e.
Proof. Žc. and Žd. are clearly equivalent. Since l p has property P1r p the implications Ž a. ª Ž b . ª Ž c . were shown in 3.2. So it remains to show N that Žc. implies Ža.. Assume Ý is1 1rpi G 1, then the Holder inequality ¨ guarantees that the canonical multiplication operator F : l p 1 = ??? = l p N ª l 1 , F Ž Ž j n1 . n , . . . , Ž j nN . n . [ Ž j n1 ??? j nN . n &
ˆ : l p mp . . . mp l p ª l 1 is well defined and onto hence its & linearization F 1 N is onto as well. It follows that l p 1mp . . . mp l p N cannot be reflexive. X 4.2. Observing that LŽ l p 1, . . . , l p N ; l q . s LŽ l p 1, . . . , l p N , l q ; K. holds isometrically, one obtains the equivalence of Ža. and Žd. of the
THEOREM.
Let pi , q gx1, `w, then the following statements are equi¨ alent:
N Ža. Ý is1 1rpi - 1rq. Žb. E¨ ery A g LŽ l p , . . . , l p ; l q . is compact. 1 N Žc. E¨ ery A g LŽ l p , . . . , l p ; l q . is sequentially weak to norm contin1 N uous. Žd. LŽ l p , . . . , l p ; l q . is reflexi¨ e. 1 N
This generalizes Pitt’s theorem for linear operators l p ª l q to multilinear operators, one might call it the Pełczynski-Pitt theorem. `
540
ALENCAR AND FLORET
Proof. Corollary 1 in 3.2 Žand 2.3. gives that Ža. ª Žc. ª Žb.. That Žb. ª Ža. follows again by using the canonical multiplication operator N Ž l p 1 = ??? = l p N ª l q Žif Ý is1 1rpi . G 1rq . which is not compact. 4.3. For p1 s ??? s pN s p, the condition on the indices is N - p Žin 4.1. and Nq - p Žin the Pełczynski-Pitt theorem.. It is clear Žby what ` was said in 3.2 and by the fact that l p ( l pN . that one has equivalent conditions on polynomials as stated in corollary 3 of 3.2.
5. TYPE, COTYPE AND THE BANACH-SAKS PROPERTY 5.1. Up to now the main applications of the results in section 3 were about sequence spaces l p . Moreover there are many obvious applications if the Ei are subspaces of c 0 Žhence have P0 . and F has some rank. Spaces of ŽRademacher. type and cotype Žsee w10x or w25x for the definition. can also be used, since the following result holds. THEOREM.
Let 1 - p F 2 F q - `.
Ž1. E¨ ery Banach space of type p has property P1r p . Ž2. E¨ ery Banach space E of cotype q has loose rank 1rq. E¨ en more: e¨ ery seminormalized weakly null sequence in E has a subsequence Ž x n . satisfying c < B < 1r q F
Ý
xn
ngB
for some c ) 0 and all finite B ; N. The proof will use the technique of spreading models which was already successfully applied to this kind of problems by Rakov w27, 28x, Farmer w14x, Farmer and Johnson w15x, and Gonzalo and Jaramillo w17x. In particular, Rakov showed in w28x the result Ž1. for the formally weaker Banach-Saks type p Žbut see 5.8 below. and Gonzalo and Jaramillo w17x have the related theorem for upper and lower estimates. 5.2. Spreading models allow to choose subsequences which are nearly unconditionally basic sequences. LEMMA 1. Let Ž e n . be a seminormalized unconditional basic sequence in a Banach space F. If F has type p then Ž e n . is t 1r p-con¨ ergent to 0. If F has cotype q then there is a constant c such that < B < 1r q F c 5Ý ng B e n 5 for all finite B ; N. Ž1. Ž2.
541
CONTINUITY OF MULTILINEAR MAPPINGS
Proof. There is a constant Žsee e.g. w10, p. 358x. such that N
1
N
Ý < ln < en
c
N
Ý ln en
F
ns1
Ý < ln < en
Fc
ns1
,
ns1
in particular, 1
Ý
c
en F
Ý rn Ž t . e n
ngB
Fc
ngB
Ý
en
ngB
for the Rademacher functions rn : w0, 1x ª y1, 14 . Therefore the definitions of type and cotype give the result. 5.3. If Ž x n . is a seminormalized s Ž E, EX .-null sequence then it has a subsequence Ž yn . with a spreading model Ž F, Ž e n .., i.e. a Banach space F with an unconditional seminormalized basis Ž e n . such that: Ža. For all « ) 0, all N g N and all l1 , . . . , l N g K there is an m g N such that N
N
Ý l k yn ks1
y
k
Ý lk en
F«
k
ks1
E
for all m F n1 - ??? - n N
F
Žb. For all n1 - n 2 - ??? - n N and scalars l k the equality N
N
Ý lk en
s
k
ks1
Ý lk ek ks1
F
. F
holds. For a proof of this, see w6x. LEMMA 2.
Let 0 - a - 1.
Ž1. If Ž e n . ta-con¨ erges to 0 then Ž yn . has a subsequence Ž z n . which ta-con¨ erges to 0. Ž2. If < B < a F c 5Ý ng B e n 5 F for some c ) 0 and all finite B ; N, then Ž yn . has a subsequence which satisfies the same type of lower estimate. It is easy to see that the converse statements in Ž1. and Ž2. hold as well. Proof. Ž1. By property Ža. of the spreading model for « s l1 s ??? s l N s 1 there is a strictly monotone function b : N ª N such that for all B ; N with < B < s N and min B G b Ž N .
Ž ).
Ý ngB 1r a x.
Define t Ž n. [ b Žw n and z n [ yt Ž n. .
F
yn E
Ý ngB
q 1 F c< B< a
en F
Žwhere w x x denotes the integral part of x g w1, `..
542
ALENCAR AND FLORET
For a given B ; N with < B < s N take m [ w N a x q 1, B0 [ w1, mwlB, B1 [ B_ B0 and choose D ; w m, `wlN_ B1 such that < D < s < B0 < F m y 1 F N a. Then
Ý
F
yt Ž n.
ngB
q
yt Ž n.
Ý ngB 0
E
q
yt Ž n.
Ý ngDjB1
E
Ý
yt Ž n.
ngD
E
E
F Ž < D < q < B0 < . sup 5 yn 5 E q cN a F ˜ cN a by Ž). since min t Ž D j B1 . G t Ž m. G b Ž N .. Ž2. The same b as above gives that
Ý
F
en
ngB
F
q1
yn
Ý ngB
E
for all B ; N with < B < s N and min B G b Ž N .. This time, the index for the subsequence will be defined by t Ž n. [ b w n2r a x. Proceeding as before but with m [ w N a r2 x q 1 hence t Ž m. G b Ž N ., we obtain c< B< a F
Ý
F
Ý
s
en
ngB
F
Ý
q
yt Ž n.
ngB
F
q
yt Ž n. E
Ý Ý
E
q1
yt Ž n.
ngB 0
q1
yt Ž n.
ngDjB1
E
q Ž < D < q < B0 < . sup 5 yn 5 E q 1.
yt Ž n.
ngB
Ý ngD
E
F
en
Ý ng t Ž DjB1 .
F
E
Since < D < s < B0 < F m y 1 F N a r2 one obtains for some constant c1
Ý
G c < B < a y c1 < B < a r2 .
yt Ž n.
ngB
E
Passing again to subsequence, one may assume that Ž yt Ž n. . is a basic Žstill seminormalized. sequence, hence there is a g ) 0 with g F 5Ý ng B yt Ž n. 5 E for all finite f / B ; N. It follows that
Ý ngB
G max g , c < B < a y c1 < B < a r2 4
yt Ž n. E
which easily implies a lower estimate of the form c 2 < B < a.
CONTINUITY OF MULTILINEAR MAPPINGS
543
5.4. Now everything is ready for the Proof of Theorem 5.1: Ž1. Let Ž x n . be a s Ž E, EX .-null seminormalized sequence with spreading model Ž F, Ž e n ... Since F is finitely represented in E Žsee w6x. it has also type p and lemmas 1 Ž1. and 2 Ž1. give the result. Ž2. Take a seminormalized ta-null sequence Ž x n . with a spreading model Ž F, Ž e n ... Since F has cotype q one obtains from Lemmas 1 Ž2. and 2 Ž2. that there is a subsequence Ž xt Ž n. . with c1 < B < 1r q F 5Ý ng B xt Ž n. 5 E . Adapting Lemma 1 Ž2., it is obvious that}as in 2.3 Ž5. }the statement Ž2. of the theorem holds also if id E is absolutely Ž q, 1.-summing. 5.5. The main theorem, for example, implies that for 1 - p F 2 F q - `, all A g LŽ L q w0, 1x, l 3 ; l 9r8 . and all a g LŽ l 5 , l 5 ; L p w0, 1x. are sequentially weak to norm continuous and compact and these spaces of operators are reflexive, since L q has type 2 and L p has cotype 2. Using the fact that l 2 ; L r w0, 1x is complemented for all 1 - r - ` the results in sections 3 and 4 imply equivalences of the kind L Ž L q w 0, 1 x , l 3 ; l s . is reflexive if and only if 1 - s - 6r5 L Ž l 5 , l s ; L p w 0, 1 x . is reflexive if and only if 10r3 - s - `, and in these cases all the operators are compact. 5.6. A Banach space of type p ) 1 has some cotype Žsee w25x, p. 39., hence one obtains the COROLLARY.
E¨ ery Banach space of type p ) 1 has positi¨ e rank.
It follows that every N-linear continuous mapping on c 0 with values in a Banach space of some type p ) 1 or of some cotype is sequentially weak to norm continuous and throws weakly compact subsets of c 0N onto norm compact sets. 5.7. Theorem 5.1 holds Žwith a weaker estimate in Ž2.. also for weak type and weak cotype Žsee w19, 20x for definitions. }this was suggested to us by M. Junge. PROPOSITION. Ž1. Ž2.
Let 1 - p - 2 F q - `.
E¨ ery Banach space of weak type p has P1r p . E¨ ery Banach space of weak cotype q has loose rank 1rq.
In particular, every Banach space of weak type 2 has Pa for all a gx1r2, 1w. Proof. It follows from w19, Theorem 3.4x Žand the fact that Rademacher-averages are smaller than Gauss-averages . that in spaces F with weak
544
ALENCAR AND FLORET
type p - 2 the inequality 1r2
2
n
žH Ý
rk Ž t . ek
ks1
dt F
/
F cn1r p
holds for all e k g F with 5 e k 5 s 1. Now one can proceed as in the proof of the theorem. The statement Ž2. follows directly from the theorem and the fact that weak cotype q implies cotype q q « for all « ) 0 Žthis follows from the results in w20x and w29, Theorem 12.2x.. 5.8. The same techniques of spreading models can be used to simplify the definition of the property Pa . In view of 2.2 Ž3. the following result improves w27, Theorem 3x, and w9, Theorem 2.3x. PROPOSITION. Let 0 - a - 1 and E be a normed space. Then E has property Pa if and only if E is of Banach-Saks type 1ra , i.e.: e¨ ery seminormalized s Ž E, EX .-null sequence admits a subsequence Ž x n . such that N
F cN a
xn
Ý ns1
E
for some c G 0 and all N g N. Proof. Assume that the condition is satisfied, take a seminormalized s Ž E, EX .-null sequence Ž x n . with spreading model Ž F, Ž e n .., choose a strictly monotone function b : N ª N such that y
xn
Ý ngB
for all B ; b Ž n . , b Ž n . q 1, . . . 4
F1
en
Ý ngB
E
F
with < B < s N and a subsequence Ž xt Ž n. . with t Ž n. G b Žw n1r a x. and N 5Ý ns1 xt Ž n. 5 E F cN a. Then, as in the proof of Ž2. in Lemma 2 with B s 1, . . . , N 4 , one obtains N
N
F
Ý en ns1
Ý ns1
F
q Ž < D < q < B0 < . sup 5 x n 5 E q 1 F ˜ cN a .
xt Ž n. E
Now the calculation in the proof of Ž1. in Lemma 2 and property Žb. of the spreading model give that
Ý ngB
F Ž < B0 < q < D < . sup 5 x n 5 E q
xt Ž n.
Ý ngDjB1
E
F c1 N a q
Ý
s c1 N a q
F
N
q 1 F Ž c1 q ˜ c q 1. N a .
Ý eN ns1
E
q1
en
ngDjB1
xt Ž n.
F
CONTINUITY OF MULTILINEAR MAPPINGS
545
5.9. It follows from w18x that for 0 - a - 1 an Orlicz sequence space h M not containing l 1 has property Pa if and only if the Orlicz function M satisfies: sup
½
M Ž st . M Ž s . t 1r a
: s, t g x 0, 1 x - `
5
Other conditions for Orlicz sequence spaces having property Pa appeared in w28x.
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