Multiple Rademacher means and their applications

J. Math. Anal. Appl. 386 (2012) 699–708

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Multiple Rademacher means and their applications Dumitru Popa Dept. of Math., Ovidius Univ. of Constanta, Bd. Mamaia 124, 8700 Constanta, Romania

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 21 February 2011 Available online 17 August 2011 Submitted by Richard M. Aron

We evaluate multiple Rademacher means for various classes of multilinear operators. Under the assumption that the range has cotype q, these results show that the class of multiple (q, 2)-summing operators plays an important role. Further, we get new extensions of Bu’s theorem in the multilinear case. © 2011 Elsevier Inc. All rights reserved.

Keywords: Absolutely p-summing multilinear operator Cotype

1. Introduction and notation The role of absolutely summing linear operators in Banach space theory is well established as the reader can see in the famous monographs of A. Pietsch, G. Pisier, N. Tomczak-Jagermann, A. Defant and K. Floret, J. Diestel, H. Jarchow and A. Tonge, see [20,19,23,7,9]. Following the pioneering work of A. Pietsch [21], in the last decade several classes of multilinear maps have been investigated as extensions of the linear concept of absolutely summing operators, see [1–3,5,8,12–18,22] and the references therein. The main purpose of this paper is to see how look the results proven in [22] for multiple summing operators in case of other classes of multilinear operators. Our results show that the class of multiple (q, 2)-summing operators plays an important role. In the proofs of these results, just like in the linear case, where Rademacher means play a key role (see [7,9,19,20,23]), the multiple Rademacher means play a key role in the multilinear case. Throughout this paper we will denote by X , X 1 , etc. Banach spaces over K = R or C, L ( X 1 , . . . , X n ; Y ) denote the Banach space of all bounded n-linear operators U : X 1 × · · · × X n → Y . We use standard notations and notions from Banach space theory, as presented e.g. in [7,9,19,20,23]. We recall some definitions and notations. Given 1  p < ∞, a Banach space X , for a finite system (xi )1i n ⊂ X we define

 w p (xi | 1  i  n) = sup

x∗ 1

n   ∗  x (xi ) p

 1p .

i =1

If 1 < p < ∞ we denote by p ∗ the conjugate of p i.e. 1p + p1∗ = 1. Let 1  p  q < ∞. A bounded linear operator U : X → Y is called (q, p )-summing, if there exists a constant C  0 such that for every choice of system (xi )1i n ⊂ X the following relation holds



n    U (xi )q

 1q  C w p ( xi | 1  i  n )

i =1

E-mail address: [email protected]. 0022-247X/$ – see front matter doi:10.1016/j.jmaa.2011.08.027

©

2011 Elsevier Inc. All rights reserved.

700

D. Popa / J. Math. Anal. Appl. 386 (2012) 699–708

and the (q, p )-summing norm of U is Πq, p (U ) = inf{C | C as above}. We denote by Πq, p ( X , Y ) the class of all (q, p )summing operators from X into Y . In case when 1  p = q < ∞ we write Π p instead of Π p , p , see [7,9,19,20,23]. Let 1 < p < ∞. A bounded linear operator U : X → Y is called Cohen p-summing, if there exists a constant C  0 such that for every choice of system (xi )1i n ⊂ X , ( y ∗i )1i n ⊂ Y ∗ the following relation holds

 n  1p n     

p  U (xi ), y ∗   C  xi  w p ∗ y ∗i  1  i  n i i =1

i =1

and the Cohen p-summing norm of U is d p (U ) = inf{C | C as above}, see [6]. In the multilinear case there are many natural extensions of the concept of summing operator. We recall those classes which we will use in this paper. The class of multiple summing operators was introduced, independently, by F. Bombal, D. Pérez-García and I. Villanueva in [3] and M. Matos in [12]. Let 1  p 1 , . . . , pn  q < ∞. A bounded n-linear operator U : X 1 × · · · × X n → Y is called multiple (q, p 1 , . . . , pn )j summing, if there exists a constant C  0 such that for every choice of systems (xi )1i j m j ⊂ X j (1  j  n) the following j

relation holds

m

 

 U x1 , . . . , xn q

1 ,...,mn

i1

i 1 ,...,in =1

 1q

in





 C w p 1 x1i 1  1  i 1  m1 · · · w pn xnin  1  in  mm

mult and the multiple (q, p 1 , . . . , pn )-summing norm of U is Πqmult , p 1 ,..., pn (U ) = inf{C | C as above}. We denote by Πq, p 1 ,..., pn ( X 1 , . . . , X n ; Y ) the class of all multiple (q, p 1 , . . . , pn )-summing operators from X 1 × · · · × X n into Y . In case when 1  p 1 = · · · = mult pn = p  q < ∞ we write Πqmult , p instead of Πq, p ,..., p .

Let 1  p 1 , . . . , pn < ∞ and define t ∈ (0, ∞) by

1 t

=

1 p1

+ ··· +

1 pn

. A bounded n-linear operator U : X 1 × · · · × X n → Y j

is called ( p 1 , . . . , pn )-dominated if there exists C > 0 such that for each (xi )1i m ⊂ X j (1  j  n) the following holds



 m   1

 t 



U x , . . . , xn t  C w p 1 x1i  1  i  m · · · w pn xni  1  i  m i i 1

i =1

and

δ p 1 ,..., pn (U ) = inf{C | C as above}. We denote by δ p 1 ,..., pn ( X 1 , . . . , X n ; Y ) the class of all ( p 1 , . . . , pn )-dominated operators from X 1 ×· · ·× X n into Y , see [10]. Let 1  p < ∞. A bounded n-linear operator U : X 1 × · · · × X n → Y is called p-semi-integral if there exists C > 0 such j that for each (xi )1i m ⊂ X j (1  j  n) the following holds



m   1

 U x , . . . , xn  p i

i

 1p

 C

i =1

sup

(x∗1 ,...,xn∗ )∈ B X ∗ ×···× B X ∗ n

1

m   ∗ 1

 x x · · · x∗ xn  p 1

i

n

 1p

i

i =1

and

J psemi (U ) = inf{C | C as above}. We denote by J psemi ( X 1 , . . . , X n ; Y ) the class of all p-semi-integral operators from X 1 × · · · × X n into Y , see [5]. The concept introduced by J.S. Cohen in the linear case, was extended recently to the multilinear case, see [1,2,13]. Let 1 < p < ∞. A bounded n-linear operator U : X 1 × · · · × X n → Y is called Cohen strongly p-summing if there exists j C > 0 such that for each (xi )1i m ⊂ X j (1  j  n), each ( y ∗i )1i m ⊂ Y ∗ the following holds m   1

  U x , . . . , xn , y ∗   C i

i

i

i =1



m   1p  np x  · · · x  i

i

 1p





w p ∗ y ∗i  1  i  m



i =1

and

d p (U ) = inf{C | C as above}. We denote by d p ( X 1 , . . . , X n ; Y ) the class of all Cohen strongly p-summing operators from X 1 × · · · × X n into Y .

D. Popa / J. Math. Anal. Appl. 386 (2012) 699–708

By (rn )n∈N we denote the sequence of Rademacher functions. Recall that, if 0 < p < ∞, Z a Banach space and ( zi )1i n ⊂ Z , then, the Rademacher means, defined by

701

ρ p (zi | 1  i  n) are

p  1  1  n p     ρ p ( z i | 1  i  n) = r i (t ) zi  dt ,    0

i =1

see [7,9,19,20,23]. Khinchin’s inequality. If 0 < p < ∞, there are positive constants A p , B p , called Khinchin’s constants, such that

 Ap

n 

 12 |ai |

2

i =1

p  1  1  n  n  12 p      2  ai r i (t ) dt  Bp |ai |    0

i =1

i =1

for each choice of scalars a1 , . . . , an ; see [7,9]. Kahane–Khinchin’s inequality. If 0 < p , q < ∞, there exists a constant K p ,q > 0, called Kahane–Khinchin’s constant, such that

q  1 p  1  1  n  1  n q p         r i (t )xi  dt  K p ,q r i (t )xi  dt       i =1

0

i =1

0

i.e.

ρq (xi | 1  i  n)  K p,q ρ p (xi | 1  i  n) for each Banach space X and each choice of elements x1 , . . . , xn ∈ X ; see [9]. Let 2  q < ∞. A Banach space X is said to have cotype q if there exists a constant C  0 such that for each elements x1 , . . . , xn ∈ X we have



n 

 1q xi q

 C ρ2 (xi | 1  i  n)

i =1

and the cotype q constant of X is denoted by C q ( X ) = inf{C | C as above}. When we say that a Banach space has cotype q we always understand 2  q < ∞, see [7,9,19,20,23]. m ,...,m For 0 < p < ∞, Z a Banach space, n a natural number, ( zi 1 ···in )i 1,...,i =n1 ⊂ Z we define multiple Rademacher means by 1



ρ p (zi1 ···in | 1  i 1  m1 , . . . , 1  in  mn ) =

n

p  1p  ,...,mn  m1    ···  r i 1 (t 1 ) · · · r in (tn ) zi 1 ···in  dt 1 · · · dtn ,  

[0,1]n

i 1 ,...,in =1

see also [15] where this concept is attributed to M. Matos. We recall Multiple Khinchin’s inequality. If 0 < p < ∞, then

 [ A p ]n





 12

m1 ,...,mn

|ai 1 ···in |2



i 1 ,...,in =1

p  1p  m1 ,...,mn    ···  r i 1 (t 1 ) · · · r in (tn )ai 1 ···in  dt 1 · · · dtn  

[0,1]n

 n

 [B p ]

i 1 ,...,in =1



 12

m1 ,...,mn

|ai 1 ···in |

2

i 1 ,...,in =1 m ,...,m

for each choice of scalars (ai 1 ···in )i 1,...,i =n1 ; A p , B p are Khinchin’s constants; see [7, p. 455], or [11, Chapter IX, Problem 3.6]. n 1

702

D. Popa / J. Math. Anal. Appl. 386 (2012) 699–708

Multiple Kahane–Khinchin’s inequality. If 0 < p , q < ∞ then

ρq (zi1 ···in | 1  i 1  m1 , . . . , 1  in  mn )  [ K p,q ]n ρ p (zi1 ···in | 1  i 1  m1 , . . . , 1  in  mn ) m ,...,m

for each Banach space Z and each choice of elements ( zi 1 ···in )i 1,...,i =n1 ⊂ Z ; K p ,q are Kahane–Khinchin’s constants; see n 1 [1, Proposition 3.1]. 2. The results We recall a multiple cotype inequality, see [8, Lemma 2.2], [16, Proposition 3.8], [18, Lemma 3.9], [22, Lemma 3], which was the starting-point for our paper. m ,...,m

Lemma 1. If Y has cotype q, 1  r < ∞ and ( y i 1 ···in )i 1,...,i =n1 ⊂ Y , then n 1

m



 1q

1 ,...,mn

q

 y i 1 ···in 

n

 C q (Y ) K r ,2 ρr ( y i 1 ···in | 1  i 1  m1 , . . . , 1  in  mn )

i 1 ,...,in =1

where K r ,2 is the Kahane–Khinchin constant. We further need the following simple observation. Let U : X 1 × · · · × X n → Y be a bounded n-linear operator and j (xi j )1i j m j ⊂ X j (1  j  n). From the definition of multiple Rademacher means and separate linearity of U in each variable, we get the following equality







ρ p U x1i1 , . . . , xnin  1  i 1  m1 , . . . , 1  in  mn =







 U x1 (t 1 ), . . . , xn (tn )  p dt 1 · · · dtn

···

 1p

(∗)

[0,1]n j

j

j

where x j (t j ) = r1 (t j )x1 + r2 (t j )x2 + · · · + rm j (t j )xm j (1  j  n); 0 < p < ∞. 2.1. The case of ( p 1 , . . . , pn )-dominated operators Lemma 2. Let X be a Banach space, μ a regular Borel probability measure on Ω = B X ∗ endowed to the weak∗ -topology and 1  p < ∞. Then for each x1 , . . . , xm ∈ X we have

1  0

 

r1 (t )x∗ (x1 ) + · · · + rm (t )x∗ (xm ) p dμ x∗

 1p

dt  B p w 2 (xi | 1  i  m).

Ω



1

Proof. Define x : [0, 1] → X by x(t ) = r1 (t )x1 + · · · + rm (t )xm and h : [0, 1] → [0, ∞) by h(t ) = ( Ω |x∗ (x(t ))| p dμ(x∗ )) p . Then we have

1  0

 ∗

 x x(t )  p dμ x∗

 1p

 1

1 dt =

h(t ) dt  0

Ω

 1 =

 1

 1p p

=

h (t ) dy 0

 ∗

 x x(t )  p dt dμ x∗

 1p

 ∗

 x x(t )  p dμ x∗ dt

0 Ω

Ω 0

in the last equality we used Fubini’s theorem. If x∗ ∈ B X ∗ , from Khinchin’s inequality we get

1 0

 ∗

 x x(t )  p dt =

1

  r1 (t )x∗ (x1 ) + · · · + x∗ (xm )rm (t ) p dt

0

 [B p ]p

 sup

x∗ 1

   1 p  ∗ x (x1 )2 + · · · + x∗ (xm )2 2 .

The statement of the lemma follows from these relations and that

μ is a probability. 2

 1p

D. Popa / J. Math. Anal. Appl. 386 (2012) 699–708

703 j

Proposition 3. Let 1  p 1 , . . . , pn < ∞ and U ∈ δ p 1 ,..., pn ( X 1 , . . . , X n ; Y ). Then for each (xi )1i j m j ⊂ X j (1  j  n) we have j



 1  i 1  m1 , . . . , 1  in  mn ρ1 U 



 B p 1 · · · B pn δ p 1 ,..., pn (U ) w 2 x1i 1  1  i 1  m1 · · · w 2 xnin  1  in  mn .

x1i 1 , . . . , xnin

Proof. From (∗) formula we have







ρ1 U x1i1 , . . . , xnin  1  i 1  m1 , . . . , 1  in  mn =





 · · · U x1 (t 1 ), . . . , xn (tn )  dt 1 · · · dtn

[0,1]n j

j

j

where x j (t j ) = r1 (t j )x1 + r2 (t j )x2 + · · · + rm j (t j )xm j , 1  j  m j . Since U is ( p 1 , . . . , pn )-dominated, by the domination theorem, see [10], there exist regular Borel probability measures μ j on Ω j = B X ∗ (1  j  n) such that for each (x1 , . . . , xn ) ∈ X 1 × · · · × Xn we have j

  U (x1 , . . . , xn )  δ p

 (U ) 1 ,..., pn

 ∗ 

x (x1 ) p 1 dμ1 x∗ 1

 p1

1

1

 ···

 ∗ 

x (xn ) pn dμn x∗ n

Ω1

 p1

n

.

n

Ωn

Then





ρ1 U x1i1 , . . . , xnin  1  i 1  m1 , . . . , 1  in  mn 



 δ p 1 ,..., pn (U )

···

 ∗

 x x1 (t 1 )  p 1 dμ1 x∗ 1

[0,1]n

 1  = δ p 1 ,..., pn (U )

 p1

1

Ω1

 ∗

 x x1 (t 1 )  p 1 dμ1 x∗ 1

0



 p1

1

1

1



 ∗

 x xn (tn )  pn dμn x∗

··· 

n

n

n

Ωn

 1 

dt 1 · · · 0

Ω1

 p1

dt 1 · · · dtn

 1  ∗ ∗ pn

 pn x xn (tn )  dμn x dtn n n

(1)

Ωn

in the last equality we used Fubini’s theorem. The statement of the proposition follows from the relation (1) and Lemma 2.

2

The next result is an extension of Theorem 3.10 in [18]. Theorem 4. Suppose that Y has cotype q. Then, for any Banach spaces X 1 , . . . , X n and all 1  p 1 , . . . , pn < ∞, we have

δ p 1 ,..., pn ( X 1 , . . . , Xn ; Y ) ⊂ Πqmult ,2 ( X 1 , . . . , X n ; Y ). j

Proof. Let U ∈ δ p 1 ,..., pn ( X 1 , . . . , X n ; Y ) and let (xi )1i j m j ⊂ X j (1  j  n). Since Y has cotype q from Lemma 1 we have j

m

 

 U x1 , . . . , xn q

1 ,...,mn

i 1 ,...,in =1

i1

 1q

in

n



 C q (Y ) K 1,2 ρ1 U x1i 1 , . . . , xnin  1  i 1  m1 , . . . , 1  in  mn .

(1)

From Proposition 3 and (1) we get

m

 

 U x1 , . . . , xn q

1 ,...,mn

i 1 ,...,in =1

i1

 1q

in

n 



 C q (Y ) K 1,2 B p 1 · · · B pn δ p 1 ,..., pn (U ) w 2 x1i 1  1  i 1  m1 · · · w 2 xnin  1  in  mn

i.e. U ∈ Πqmult ,2 ( X 1 , . . . , X n ; Y ) and

n πqmult ,2 (U )  [C q (Y ) K 1,2 ] B p 1 · · · B pn δ p 1 ,..., pn (U ). 2

2.2. The case of p-semi-integral operators j

Proposition 5. Let 1  p < ∞ and U ∈ J psemi ( X 1 , . . . , X n ; Y ). Then for each (xi )1i j m j ⊂ X j (1  j  n) we have j



















ρ p U x1i1 , . . . , xnin  1  i 1  m1 , . . . , 1  in  mn  [ B p ]n J psemi (U ) w 2 x1i1  1  i 1  m1 · · · w 2 xnin  1  in  mn .

704

D. Popa / J. Math. Anal. Appl. 386 (2012) 699–708

Proof. Since U is p-semi-integral, by the domination theorem, see [5], there exist a regular Borel probability measure Ω1 × · · · × Ωn (Ω j = B X ∗ , 1  j  n) such that for each (x1 , . . . , xn ) ∈ X 1 × · · · × Xn we have

μ on

j

   U (x1 , . . . , xn )  J semi (U )



 ∗ 

x (x1 ) · · · x∗ (xn ) p dμ x∗ , . . . , x∗

p

n

1

1

 1p .

n

(1)

Ω1 ×···×Ωn

Then, from (∗) formula and (1) we get





ρ p U x1i1 , . . . , xnin  1  i 1  m1 , . . . , 1  in  mn 





J psemi (U )

  1p  ∗



x x1 (t 1 ) · · · x∗ xn (tn )  p dμ x∗ , . . . , x∗ dt 1 · · · dtn n n 1 1



···

 = J psemi (U )



Ω1 ×···×Ωn

[0,1]n





Ω1 ×···×Ωn

 1p 



 p · · · x∗1 x1 (t 1 ) · · · xn∗ xn (tn )  dμ x∗1 , . . . , xn∗ dt 1 · · · dtn

(2)

[0,1]n

in the last equality we used Fubini’s theorem. Now let (x∗1 , . . . , xn∗ ) ∈ Ω1 × · · · × Ωn . Then, by Fubini’s theorem we have







 p · · · x∗1 x1 (t 1 ) · · · xn∗ xn (tn )  dt 1 · · · dtn =

[0,1]n

 1

  1   ∗  ∗

 p

 p x x1 (t 1 )  dt 1 · · · x xn (tn )  dtn . n 1

0

(3)

0

Since, from the inequality of Hincin, we get, for example,

1

  ∗   1  p

   x x1 (t 1 )  p dt 1  [ B p ] p sup x∗ x1 2 + · · · + x∗ x1 2 2 m1 1 1

(4)

x∗ 1

0

the statement of the proposition follows from the relations (2), (3), (4) and that

μ is a probability. 2

Theorem 6. Suppose that Y has cotype q. Then, for any Banach spaces X 1 , . . . , X n and all 1  p < ∞, we have

J psemi ( X 1 , . . . , X n ; Y ) ⊂ Πqmult ,2 ( X 1 , . . . , X n ; Y ). j

Proof. Let U ∈ J psemi ( X 1 , . . . , X n ; Y ) and let (xi )1i j m j ⊂ X j (1  j  n). Since Y has cotype q from Lemma 1 we have j

m

 

 U x1 , . . . , xn q

1 ,...,mn

i 1 ,...,in =1

i1

 1q

in

n



 C q (Y ) K p ,2 ρ p U x1i 1 , . . . , xnin  1  i 1  m1 , . . . , 1  in  mn .

(1)

Then, from (1) and Proposition 5 we get

m

 

 U x1 , . . . , xn q

1 ,...,mn

i 1 ,...,in =1

i1

in

i.e. U ∈ Πqmult ,2 ( X 1 , . . . , X n ; Y ) and

 1q

n 



 C q (Y ) K p ,2 B p J psemi (U ) w 2 x1i 1  1  i 1  m1 · · · w 2 xnin  1  in  mn

n semi πqmult ( U ). 2 ,2 ( U )  [ C q ( Y ) K p ,2 B p ] J p

2.3. The case of Cohen strongly p-summing multilinear operators Let U : X 1 × · · · × X n → Y be a bounded n-linear operator. The (bounded linear) operator U ∗ : Y ∗ → L ( X 1 , . . . , X n ; K) defined by



U ∗ y∗









x1 , . . . , xn = U x1 , . . . , xn , y ∗



is called the dual of U . We further need the following result of Saadi and Mezrag, see [13, Theorem 3.4], if 1 < p < ∞, then a bounded n-linear operator U : X 1 × · · · × X n → Y is Cohen strongly p-summing if and only if U ∗ : Y ∗ → L ( X 1 , . . . , X n ; K) is p ∗ -summing (linear). In addition π p ∗ (U ∗ : Y ∗ → L ( X 1 , . . . , X n ; K)) = d p (U ).

D. Popa / J. Math. Anal. Appl. 386 (2012) 699–708

705

The proof of the following proposition is analogous to that of Proposition 2 in [22]. We omit its proof. j

Proposition 7. Let 1  p 1 , . . . , pn  q < ∞ and U ∈ Πqmult , p 1 ,..., pn ( X 1 , . . . , X n ; Y ). Then for each (xi )1i j m j ⊂ X j (1  j  n) we j

have





ρq U x1i1 , . . . , xnin  1  i 1  m1 , . . . , 1  in  mn



1 

n 

   B p 1 · · · B pn Πqmult , p 1 ,..., pn (U ) w 2 xi 1 1  i 1  m1 · · · w 2 xin 1  in  mn .

In the remainder of the paper we will prove multilinear extensions of Bu’s theorem, see [4]. A first multilinear extension of Bu’s theorem was proven by D. Achour and L. Mezrag in [1] and also a polynomial version was shown in [2]. Lemma 8. Let m1 , . . . , mn be natural numbers and U : l2 1 × · · · × l2 n → Y a bounded n-linear operator. Then for each y ∗ ∈ Y ∗ we m

m

m ,...,m 1 have U ∗ ( y ∗ )  ( i 1,...,i =n1 |U (e 1i , . . . , eni ), y ∗ |2 ) 2 . n n 1 1 m

m

Proof. Indeed, if (x1 , . . . , xn ) ∈ l2 1 × · · · × l2 n , then from x j =







U x1 , . . . , xn , y ∗ =

 

m1 ,...,mn

i 1 ,...,in =1





m j

i j =1

j

j

x j , e i j e i j we get





x1 , e 1i 1 · · · xn , enin U e 1i 1 , . . . , enin , y ∗



and therefore, by Cauchy–Buniakowski–Schwartz’s inequality

 1

  U x , . . . , xn , y ∗  

m

      x1 , e 1 · · · xn , en 2

1 ,...,mn

i 1 ,...,in =1

    = x1  · · · xn 

i1

m

 12  m

in

 

  U e 1 , . . . , en , y ∗ 2

1 ,...,mn

i 1 ,...,in =1

 

  U e 1 , . . . , en , y ∗ 2

1 ,...,mn

i1

i 1 ,...,in =1

i1

 12

in

 12

in

2

which concludes the proof.

Next theorem is the main step in the proofs of our multilinear variants of Bu’s theorem. Theorem 9. Let m1 , . . . , mn be natural numbers and Y a Banach space. Then: (i) For each 1  p 1 , . . . , pn  q < ∞, each 1 < p < ∞ we have

m1

m1

mn mn Πqmult , p 1 ,..., pn l2 , . . . , l2 ; Y ⊂ d p l2 , . . . , l2 ; Y . In addition



d p (U ) 

K q, p ∗

n B p 1 · · · B pn Πqmult , p 1 ,..., pn (U ).

A p∗

(ii) For each 1  p 1 , . . . , pn < ∞, each 1 < p < ∞ we have

1

m1

mn mn δ p 1 ,..., pn lm 2 , . . . , l2 ; Y ⊂ d p l2 , . . . , l2 ; Y . In addition



d p (U ) 

K 1, p ∗

n B p 1 · · · B pn δ p 1 ,..., pn (U ).

A p∗ m

m

Proof. Take U ∈ L (l2 1 , . . . , l2 n ; Y ). From Lemma 8 we have

 ∗ ∗  U y  

m

 

  U e 1 , . . . , en , y ∗ 2

1 ,...,mn

i 1 ,...,in =1

i1

in

From multiple Khinchin’s inequality we have

 12

for y ∗ ∈ Y ∗ .

(1)

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D. Popa / J. Math. Anal. Appl. 386 (2012) 699–708

m

 

  U e 1 , . . . , en , y ∗ 2

1 ,...,mn

i1

i 1 ,...,in =1



[ A p ∗ ]n

i 1 ,...,in =1

[0,1]n



1

=

in

 p∗  1∗  m1 ,...,mn p  1

∗   n r i 1 (t 1 ) · · · r in (tn ) U e i 1 , . . . , e in , y  dt 1 · · · dtn ···   



1

[ A p ∗ ]n

 12

 1∗ p   p ∗

· · ·  U x1 (t 1 ), . . . , xn (tn ) , y ∗  dt 1 · · · dtn

[0,1]n

which by (1) gives us

 ∗ ∗  U y  



1

[ A p ∗ ]n



  p ∗

· · ·  U x1 (t 1 ), . . . , xn (tn ) , y ∗  dt 1 · · · dtn



1 p∗

(2)

[0,1]n

j

j

j

where x j (t j ) = r1 (t j )e 1 + r2 (t j )e 2 + · · · + rm j (t j )em j (1  j  n). Let ( yk∗ )1km ⊂ Y ∗ . From (2) we get



m   ∗ ∗  p ∗ U y 



1 p∗



k

k =1



1

[ A p ∗ ]n

 1∗  m p 

∗  p ∗  U x1 (t 1 ), . . . , xn (tn ) , y  dt 1 · · · dtn . ···

(3)

k

k =1

[0,1]n

On the other hand, for each (t 1 , . . . , tn ) ∈ [0, 1]n obviously we have the following relationship



m    ∗

 U x1 (t 1 ), . . . , xn (tn ) , y ∗  p



1 p∗

 

  U x1 (t 1 ), . . . , xn (tn )  w p ∗ yk∗  1  k  m .

k

(4)

k =1

From (3), (4) and (∗) formula we infer



m   ∗ ∗  p ∗ U y 



1 p∗



k

k =1

[ A p ∗ ]n





1

=

[ A p ∗ ]n



1

 1∗ p  

 p ∗ · · · U x1 (t 1 ), . . . , xn (tn )  dt 1 · · · dtn w p ∗ yk∗  1  k  m

[0,1]n









ρ p∗ U e1i1 , . . . , enin  1  i 1  m1 , . . . , 1  in  mn w p∗ zk∗  1  k  m . m

(5) j

m

n 1 (i) If U ∈ Πqmult , p 1 ,..., pn (l2 , . . . , l2 ; Y ) then, by multiple Kahane–Khinchin’s inequality, Proposition 7 and w 2 (e i | 1  i j  j

m j )  1 we get











ρ p∗ U e1i1 , . . . , enin  1  i 1  m1 , . . . , 1  in  mn  [ K q, p∗ ]n ρq U e1i1 , . . . , enin  1  i 1  m1 , . . . , 1  in  mn  [ K q, p ∗ ]n B p 1 · · · B pn Πqmult , p 1 ,..., pn (U ).

(6)

From (5) and (6) we deduce



m   ∗ ∗  p ∗ U y 





k

k =1

Thus U ∗

π



1 p∗

K q, p ∗

n

A p∗







∗ B p 1 · · · B pn Πqmult , p 1 ,..., pn (U ) w p ∗ yk 1  k  m .

mn 1 ∗ : Y ∗ → L (lm 2 , . . . , l2 ; K) is p -summing (linear) and

p∗



∗

∗

U :Y →L

m1



m l2 , . . . , l2 n ; K

 

K q, p ∗ A p∗

n B p 1 · · · B pn Πqmult , p 1 ,..., pn (U )

which, by quoted Saadi–Mezrag’s theorem, will gives us that U is Cohen strongly p-summing and

 d p (U ) 

K q, p ∗ A p∗

n B p 1 · · · B pn Πqmult , p 1 ,..., pn (U ).

(ii) Is the same as in (i), just that in this case, instead of Proposition 7, we use Proposition 3.



2

D. Popa / J. Math. Anal. Appl. 386 (2012) 699–708

707

Theorem 10. Let H 1 , . . . , H n be Hilbert spaces and Y a Banach space. Then: (i) For each 1  p 1 , . . . , pn  q < ∞, each 1 < p < ∞ we have

Πqmult , p 1 ,..., pn ( H 1 , . . . , H n ; Y ) ⊂ d p ( H 1 , . . . , H n ; Y ). (ii) For each 1  p 1 , . . . , pn < ∞, each 1 < p < ∞ we have

δ p 1 ,..., pn ( H 1 , . . . , H n ; Y ) ⊂ d p ( H 1 , . . . , H n ; Y ). Proof. (i) Let (xi )1i m ⊂ H j (1  j  n) and ( y ∗i )1i m ⊂ Y ∗ . Then G j = Span{x1 , . . . , xm } is isometric and isomorphic with j

j

mj l2

j

mj G j → l2

and let V j : be such an isometry. Let also P j : H j → G j be the orthogonal projection, J j : G j → H j the some canonical inclusion and the diagrams m

m

l2 1 × · · · × l2 n

( V 1−1 ,..., V n−1 )

( J 1 ,..., J n )

G 1 × · · · × Gn

→

→

U

H 1 × · · · × Hn → Y .

−1 n 1 −1 Since U ∈ Πqmult , p 1 ,..., pn ( H 1 , . . . , H n ; Y ) it follows that U ◦ ( J 1 , . . . , J n ) ◦ ( V 1 , . . . , V n ) : l2 × · · · × l2 → Y is also multiple (q, p 1 , . . . , pn )-summing and m

m

  −1  

−1 −1  · · ·  J n  V −1  = Π mult   Πqmult Πqmult n q, p 1 ,..., pn (U ). , p 1 ,..., pn U ◦ ( J 1 , . . . , J n ) ◦ V 1 , . . . , V n , p 1 ,..., pn (U ) J 1  V 1 mj

1 Consider xi = P j (xi ) = V − (ai ), ai ∈ l2 . From Theorem 9(i) we have j j

j

j

j

m   −1 1



∗   U J1 V , yi  ai , . . . , J n V n−1 ani 1 i =1





n

K q, p ∗ A p∗

 ×



m   1p  np a  · · · a  i

 1p

i





n

K q, p ∗





w p ∗ y ∗i  1  i  m

i =1





−1 −1 B p 1 · · · B pn Πqmult , p 1 ,..., pn U ◦ ( J 1 , . . . , J n ) ◦ V 1 , . . . , V n

B p 1 · · · B pn Πqmult , p 1 ,..., pn (U )

A p∗



m   1p  np a  · · · a  i



i

 1p





w p ∗ y ∗i  1  i  m



i =1

i.e.

 m 1 n  m    p  p p  1

 

∗ K q , p  U x , . . . , xn , y ∗   a1  · · · an  B p 1 · · · B pn Πqmult w p ∗ y ∗i  1  i  m . , p 1 ,..., pn (U ) i i i i i A p∗

i =1

From

j ai 

−1

= V j

j (ai )

=

j  xi 

i =1

we get

 m 1 n  m    p  p p  1

 

∗ mult 1 n  U x , . . . , xn , y ∗   K q, p   B p 1 · · · B pn Πq, p 1 ,..., pn (U ) xi · · · xi w p ∗ y ∗i  1  i  m i i i A p∗

i =1

i =1

which means that U is Cohen strongly p-summing and

 d p (U ) 

K q, p ∗ A p∗

n B p 1 · · · B pn Πqmult , p 1 ,..., pn (U ).

(ii) Is analogous to that of (i), just that instead of (i) in Theorem 9 we use (ii) from that theorem.

2

Acknowledgment We would like to express our gratitude to the referee for many valuable comments, suggestions and showing us Ref. [2] which have improved the final version of the paper.

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D. Popa / J. Math. Anal. Appl. 386 (2012) 699–708

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