Hilbert–Schmidt and multiple summing operators

Collect. Math. (2012) 63:181–194DOI 10.1007/s13348-010-0028-2

Hilbert–Schmidt and multiple summing operators

Gabriela Badea · Dumitru Popa

Received: 12 May 2010 / Accepted: 15 October 2010 / Published online: 3 November 2010© Universitat de Barcelona 2010

Abstract We characterize the multiple p-summing operators for p ≥ 2, in the situationwhere the range and some of the factors of the cartesian product are Hilbert spaces. As aconsequence, we complete some recent results obtained in this area. Thus, we prove a coin-cidence result for multiple p-summing operators for p ≥ 2, which has no linear analogue.Also, we give the necessary and sufficient conditions for some concrete bilinear operators tobe multiple p-summing, p ≥ 1.

Keywords Multiple p-summing operator · Hilbert–Schmidt operator ·Coincidence theorems

Mathematics Subject Classification (1991) Primary 47H60; secondary 46B25

1 Introduction and notation

The theory of multilinear summing operators, as it was first outlined by Pietsch in 1983 [27],has developed intense lately, several interesting results being obtained in this field. One ofthe motivations, which led to this development, is finding that concept which extends best thelinear summing properties to the multilinear case. Thus, many concepts have been definedin order to extend the linear p-summing operators, such as absolutely p-summing operators,p-dominated and strongly p-summing ones, see [7,25].

Another important generalization of the linear summing concept is the multiplep-summing operator. This concept was introduced by Matos in [17], who named it “fullyabsolutely summing”, and, independently, by Bombal, Pérez-García and Villanueva in [3],

G. BadeaFaculty of Civil Engineering, University of Constanta, str.Unirii 22B, 8700 Constanta, Romaniae-mail: [email protected]

D. Popa (B)Department of Mathematics, University of Constanta, Bd.Mamaia 124, 8700 Constanta, Romaniae-mail: [email protected]

123

182 G. Badea, D. Popa

who named it ”multiple summing”, although the origin of this class goes back to [31]. Theimportance of this type of operators is also given by the fact that it has a connection withthe classical papers of Bohnenblust-Hille [2] (published in Annals of Math.) and Littlewood[16]. There are several results proven for the class of multiple summing operators, show-ing that this class seems to be one of the most successful extensions of the linear case, see[1,3,6,12,17,18,21–24,29,30] and the references therein. Also, recently several applicationshave been obtained which have reinforced the relevance of the subject ([9–11]).

For the case of Hilbert spaces, the concept of Hilbert–Schmidt multilinear operators isanother nice extension of the linear theory. This concept has been studied and used in,for example, [17,23].

The relationship between the last two concepts has been studied by several authors, amongthem, Matos in [17] and later, Pérez-García in [23], proved that every Hilbert–Schmidt multi-linear operator is multiple p-summing for every 1 ≤ p < ∞, which is the multilinear versionof the classical characterization of Hilbert–Schmidt linear operators given by Pełczynski in[20].

For more results and other related details of the multilinear theory, we recommend thebooks [14,19].

In this paper, we characterize the multiple p-summing operators for p ≥ 2, in the situationwhere the range and some of the factors of the cartesian product are Hilbert spaces. As a con-sequence, we are able to prove some coincidence results, i.e., a situation in which the spaceof multiple p-summing and multiple q-summing multilinear operators from X1 ×· · ·× Xn toY coincide for some spaces X1, . . . , Xn, Y . One of these results is related to the coincidenceresults proved by the second named author in [29, Theorem 6, Theorem 10] and indepen-dently by Botelho and Pellegrino in [4,5]. The other result gives a coincidence result for themultiple p-summing operators, p ≥ 2, which has no linear analogue.

Also, we give the necessary and sufficient conditions for some concrete bounded bilinearoperators to be multiple p-summing, p ≥ 1.

Now we fix some notation and terminology. Throughout this paper X1, . . . , Xn, Y etc.denote Banach spaces and H1, . . . , Hn, H denote Hilbert spaces. For X Banach space, BX

denotes the closed unit ball of X and X∗ will be the topological dual of X . For 1 ≤ p < ∞,

we denote by p∗ the conjugate of p i.e. 1p + 1

p∗ = 1; if p = 1, we take p∗ = ∞ and we

consider 1∞ = 0.

For k ∈ N, 1 ≤ p < ∞, we consider the Banach space lkp (X) , endowed with the norm

‖(x1, . . . , xk)‖p = (‖x1‖p + · · · + ‖xk‖p)1p . For H, a Hilbert space, lk

2(H) is also a Hilbertspace with respect to the scalar product

〈(y1, . . . , yk) , (w1, . . . , wk)〉 = 〈y1, w1〉 + · · · + 〈yk, wk〉.The notations and terminology used along the paper are standard in Banach space theory, asfor instance in [8,13,26,28,32]. These excellent books are also our main references for basicfacts, definitions and all other unexplained notion and notation.

Definition 1 Let H1, . . . , Hn, H be Hilbert spaces. A bounded multilinear operator T :H1×· · ·×Hn → H is said to be Hilbert–Schmidt if there is an orthonormal basis (e j

i j)i j ∈I j ⊂

Hj (1 ≤ j ≤ n) such that

‖T ‖H S =⎛⎝ ∑

i1∈I1,...,in∈In

∥∥T(e1

i1, . . . , en

in

)∥∥2

⎞⎠

12

< ∞.

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Hilbert–Schmidt and multiple summing operators 183

The set of all Hilbert–Schmidt operators between H1 × · · · × Hn and H is denoted byH S(H1, . . . , Hn; H) and the Hilbert–Schmidt norm of T , by ‖T ‖H S .

We should notice that the definition does not depend on the orthonormal basis that wechoose [17, Proposition 5.1]. It is easy to see that the class of Hilbert–Schmidt multilinearoperators H S(H1, . . . , Hn; H), with its norm ‖ · ‖H S, is a Hilbert space. The scalar productis given by

〈T, S〉 =∑

i1∈I1,...,in∈In

⟨T(e1

i1, . . . , en

in

), S(e1

i1, . . . , en

in

)⟩.

For a finite system (xi )1≤i≤m ⊂ X and 1 ≤ p < ∞, we denote by

wp (xi | 1 ≤ i ≤ m) = sup

⎧⎨⎩

(m∑

i=1

∣∣x∗ (xi )∣∣p

) 1p

| x∗ ∈ BX∗

⎫⎬⎭

or, when more precision is needed, wp(xi | 1 ≤ i ≤ m; X).

Definition 2 Let 1 ≤ p < ∞. A bounded linear operator T : X → Y is p-summing if thereexists a constant C > 0 such that, for every choice of a finite system (xi )1≤i≤m ⊂ X thefollowing relation holds

(m∑

i=1

‖T (xi )‖p

) 1p

≤ Cwp (xi | 1 ≤ i ≤ m).

In this case, we define the p−summing norm of T by πp(T ) = min{C : C as above } andwe denote by �p(X, Y ) the class of p-summing linear operators.

Definition 3 Let 1 ≤ p < ∞. A bounded multilinear operator T : X1 × · · · × Xn → Yis multiple p-summing if there exists a constant C > 0 such that, for every choice of finitesystems (xi j )1≤i j ≤m j ⊂ X j (1 ≤ j ≤ n) the following relation holds

⎛⎝

m1,...,mn∑i1,...,in=1

∥∥T(x1

i1, . . . , xn

in

)∥∥p

⎞⎠

1p

≤ Cwp(x1

i1| 1 ≤ i1 ≤ m1

) · · · wp(xn

in| 1 ≤ in ≤ mn

).

In this case, we define the multiple p-summing norm of T by πmultp (T ) = min{C : C as

above} and we denote by �multp (X1, . . . , Xn; Y ) the class of multiple p-summing operators.

2 Main result and consequences

We begin this section by studying the relationship between a multiple p-summing operatorand its associated k-linear operator. First, we establish the notation that we will use.

Thus, for a bounded (n+k)-linear operator T : X1 ×· · ·× Xn × Xn+1 ×· · ·× Xn+k → Y,

we define the bounded k-linear operator associated to T as T : Xn+1 × · · · × Xn+k →L(X1, . . . , Xn; Y ),

T (xn+1, . . . , xn+k) (x1, . . . , xn) = T (x1, . . . , xn, xn+1, . . . , xn+k).

Such a kind of operators where used in [12]; in fact, the operator T associated to T asabove is, in the notation of [12], the operator T C , where C = {1, 2, . . . , n}.

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Now, for T : X1 × · · · × Xn × Xn+1 × · · · × Xn+k → Y and 1 ≤ p < ∞, we considerthe following assertions:

(α) T : Xn+1×· · ·×Xn+k → �multp (X1, . . . , Xn; Y ) is bounded k-linear with respect

to the multiple p-summing norm on �multp (X1, . . . , Xn; Y )

(β) T : Xn+1 × · · · × Xn+k → �multp (X1, . . . , Xn; Y ) is multiple p-summing with

respect to the multiple p-summing norm on �multp (X1, . . . , Xn; Y )

(γ ) T is multiple p-summing

and we study the implications between them.We know that the following implications always hold: (β) ⇒ (γ ) ⇒ (α). Indeed, (β) ⇒

(γ ) was proved in [17, Proposition 2.5], [24, Proposition 2.5], although it appears implicitlyin [31, proof of Proposition 3.7]. As for the implication (γ ) ⇒ (α), it is a simple conse-quence of the definition that if T : X1 × · · · × Xn × Xn+1 × · · · × Xn+k → Y is multiplep-summing, then T takes its values in �mult

p (X1, . . . , Xn; Y ) and in this case, by a standard

closed graph argument, T : Xn+1 × · · · × Xn+k → �multp (X1, . . . , Xn; Y ) is bounded

k-linear with respect to the multiple p-summing norm on �multp (X1, . . . , Xn; Y ).

In this paper we will prove that under suitable assumptions the reverse implications alsohold. We indicate next three particular situations where the reverse implication (α) ⇒ (β)and hence, (α) ⇒ (γ ) hold; easy examples show that, as it is expected, the implication (α)⇒ (γ ) is not true in general.

If a = (an)n∈N, b = (bn)n∈N are two scalar sequences we write ab = (anbn)n∈N for theirpointwise multiplication. In the same way if x1, …, xn are scalar sequences, x1 · · · xn denotesalso their pointwise multiplication.

Proposition 1 (i) Let H1, . . . , Hn, H be all Hilbert spaces. If either Xn+1, . . . , Xn+k areL∞,λn+1 , . . . , L∞,λn+k -spaces, respectively, or Xn+1, . . . , Xn+k are all GT spaces,then a bounded (n +k)-linear operator T : H1 ×· · ·× Hn × Xn+1 ×· · ·× Xn+k → His multiple 2-summing if and only if T takes its values in H S(H1, . . . , Hn; H).

(ii) Let a ∈ l∞. Then the multiplication operator Ma : l2 × · · · × l2︸︷︷︸n times

× c0 × · · · × c0︸︷︷︸k times

→

l2, Ma (x1, . . . , xn+k) = ax1 · · · xn+k is multiple 2-summing if and only if a ∈ l2.(iii) Let a ∈ l∞. Then Ma : l2 × · · · × l2︸︷︷︸

n times

× l1 × · · · × l1︸︷︷︸k times

→ l2, Ma (x1, . . . , xn+k) =

ax1 · · · xn+k is multiple 2-summing.

Proof (i) Suppose that Xn+1, . . . , Xn+k are L∞,λn+1 , . . . , L∞,λn+k -spaces, respectively.If T takes its values in H S(H1, . . . , Hn; H), then T : Xn+1 × · · · × Xn+k →H S(H1, . . . , Hn; H) is bounded k-linear. Since the class of Hilbert–Schmidt oper-ators is a Hilbert space, hence of cotype 2, by [3, Theorem 3.1] we obtain thatT : Xn+1 × · · · × Xn+k → H S(H1, . . . , Hn; H) is multiple 2-summing. Furtherby [17, Proposition 5.7], [23, Theorem 4.2] we have that H S(H1, . . . , Hn; H) =�mult

2 (H1, . . . , Hn; H). Thus T : Xn+1 × · · · × Xn+k → �mult2 (H1, . . . , Hn; H) is

multiple 2-summing and hence, because the implication (β) ⇒ (γ ) is always true, weget that T is multiple 2-summing.In the case where all spaces Xn+1, . . . , Xn+k are GT-spaces, the proof is analogousand uses multilinear extension of Grothendieck’s Theorem, see [23, Theorem 4.5].

(ii) Since c0 is a L∞-space, by (i), Ma is multiple 2-summing if and only if Ma takesits values in H S(l2, . . . , l2︸︷︷︸

n times

; l2). Let (x1, . . . , xk) ∈ c0 × · · · × c0. The operator

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Ma(x1, . . . , xk) : l2 × · · · × l2︸︷︷︸n times

→ l2 is Hilbert–Schmidt if and only if∞∑

i=1|ai |2

|〈x1, ei 〉|2 · · · |〈xk, ei 〉|2 < ∞, hence the operator Ma : c0 × · · · × c0︸︷︷︸k times

→ l2 is well

defined, which then-again by a standard closed graph argument, is equivalent to a ∈ l2.(iii) Since l1 satisfies Grothendieck’s Theorem, i.e. l1 is a GT space, by (i), Ma is multiple

2 -summing if and only if Ma takes its values in H S(l2, . . . , l2︸︷︷︸n times

; l2), which is obvious.

Our main result gives a characterization of multiple p-summing operators for p ≥ 2 inthe case where the range and some of the factors of the cartesian product are Hilbert spaces,that relates to the connection between Hilbert–Schmidt and multiple p-summing operators(see [17,23] and the references therein). This result allows us to give some consequences andto find various examples of such operators. In order to prove it, we will need the followingwell known lemma, whose proof is included for the sake of completeness.

Lemma 1 Let H1, . . . , Hn, H be Hilbert spaces, k a natural number, T1, . . . , Tk : H1 ×· · · × Hn → H be some bounded n-linear operators and let T : H1 × · · · × Hn → lk

2(H)

be defined by

T (x1, . . . , xn) = (T1 (x1, . . . , xn) , . . . , Tk (x1, . . . , xn)) .

Then T is Hilbert–Schmidt if only if all T1, . . . , Tk are Hilbert–Schmidt. Further, ‖T ‖2H S =

‖T1‖2H S + · · · + ‖Tk‖2

H S.

Proof For each orthonormal basis (ei1)i∈I1 in H1, . . . , (ein )in∈In in Hn we have∑

(i1,...,in)∈I1×···×In

∥∥T(ei1 , . . . , ein

)∥∥2lk2 (H)

=∑

(i1,...,in)∈I1×···×In

(∥∥T1(ei1 , . . . , ein

)∥∥2 + · · · + ∥∥Tk(ei1 , . . . , ein

)∥∥2)

. (1)

Then T is a Hilbert–Schmidt operator if and only if∑

(i1,...,in)∈I1×···×In‖T (ei1 , . . . , ein )‖2

< ∞ and, by (1), if and only if∑∞

(i1,...,in)∈I1×···×In‖T1(ei1 , . . . , ein )‖2 < ∞, . . . ,∑∞

(i1,...,in)∈I1×···×In‖Tk(ei1 , . . . , ein )‖2 < ∞ i.e. if and only if all T1, . . . , Tk are Hilbert–

Schmidt. From (1) and the definition of the Hilbert–Schmidt norm we obtain the equalityfrom the statement.

Now, we are able to state and prove the main result of our paper, which gives a character-ization of multiple p-summing operators for p ≥ 2 in the case where the range and some ofthe factors of the cartesian product are Hilbert spaces.

Theorem 1 Let H1, . . . , Hn, H be Hilbert spaces, Xn+1, . . . , Xn+k be Banach spaces, T :H1 × · · · × Hn × Xn+1 × · · · × Xn+k → H a bounded (n + k)-linear operator and let2 ≤ p < ∞. The following assertions are equivalent:

(i) T is multiple p-summing.(ii) T : Xn+1 × · · · × Xn+k → H S(H1, . . . , Hn; H) is multiple p-summing with respect

to the Hilbert–Schmidt norm on H S(H1, . . . , Hn; H). Further

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[Ap]n

πmultp (T ) ≤ πmult

p

(T : Xn+1 × · · · × Xn+k → H S (H1, . . . , Hn; H)

)

≤ [Bp]n

πmultp (T ),

where Ap, Bp are Khinchin’s constants.

Proof (i)⇒(ii). Let ξ j = (xn+ jin+ j

)1≤in+ j ≤mn+ j ⊂ Xn+ j be some finite systems such that

wp(xn+ jin+ j

| 1 ≤ in+ j ≤ mn+ j ) ≤ 1 for each 1 ≤ j ≤ k and let’s define the operator

Sξ1,...,ξk : H1 × · · · × Hn → lmn+1···mn+kp (H)

Sξ1,...,ξk (h1, . . . , hn) =(

T(

h1, . . . , hn, xn+1in+1

, . . . , xn+kin+k

))1≤in+1≤mn+1,...,1≤in+k≤mn+k

.

Thus

Sξ1,...,ξk =(

T(

xn+1in+1

, . . . , xn+kin+k

))1≤in+1≤mn+1,...,1≤in+k≤mn+k

.

Then from (i) and the definition of multiple p-summing operators, it follows that Sξ1,...,ξk

is multiple p-summing and

πmultp

(Sξ1,...,ξk

) ≤ πmultp (T ).

Since 2 ≤ p < ∞, we can define s by 12 = 1

p + 1s ; if p = 2, we will consider s = ∞.

Let λ = (λin+1 . . .in+k )1≤in+1≤mn+1,...,1≤in+k≤mn+k be a finite system of scalars and let’s define

the operator Mλ : lmn+1···mn+kp (H) → lmn+1···mn+k

2 (H) by

Mλ

((hin+1 . . .in+k

)1≤in+1≤mn+1,...,1≤in+k≤mn+k

)

= (λin+1 . . .in+k hin+1 . . .in+k

)1≤in+1≤mn+1,...,1≤in+k≤mn+k

.

We note that Mλ is a bounded linear operator and from Holder’s inequality, 12 = 1

p + 1s ,

we have ‖Mλ‖ ≤ ‖λ‖lmn+1 ···mn+ks

. Then, by the ideal property of the class of all multiple

p-summing operators, it follows that Mλ ◦ Sξ1,...,ξk : H1 × · · · × Hn → lmn+1···mn+k2 (H) is

multiple p-summing and πmultp (Mλ ◦ Sξ1,...,ξk ) ≤ πmult

p (Sξ1,...,ξk )‖Mλ‖, thus

πmultp

(Mλ ◦ Sξ1,...,ξk

) ≤ πmultp (T ) ‖λ‖

lmn+1 ···mn+ks

.

Since H is a Hilbert space, lmn+1···mn+k2 (H) is a Hilbert space, from [17, Proposition 5.7],

[23, Theorem 4.2], it follows that Mλ ◦ Sξ1,...,ξk is a Hilbert–Schmidt operator and

∥∥Mλ ◦ Sξ1,...,ξk

∥∥H S ≤ [

Bp]n

πmultp

(Mλ ◦ Sξ1,...,ξk

).

By Lemma 1, all operators λin+1 . . .in+k T(

xn+1in+1

, xn+2in+2

, . . . , xn+kin+k

), 1 ≤ in+1 ≤ mn+1, . . . ,

1 ≤ in+k ≤ mn+k are Hilbert–Schmidt and

mn+1,...,mn+k∑in+1,...,in+k=1

∥∥∥λin+1 . . .in+k T(

xn+1in+1

, xn+2in+2

, . . . , xn+kin+k

)∥∥∥2

H S= ∥∥Mλ ◦ Sξ1,...,ξk

∥∥2H S

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Thus, all operators T (xn+1in+1

, xn+2in+2

, . . . , xn+kin+k

) : H1 ×· · ·× Hn → H, 1 ≤ in+1 ≤ mn+1, . . . ,

1 ≤ in+k ≤ mn+k are Hilbert–Schmidt (take for this λin+1 . . .in+k = 1) and

⎛⎝

mn+1,...,mn+k∑in+1,...,in+k=1

∣∣λin+1 . . .in+k

∣∣2 ∥∥∥T(

xn+1in+1

, xn+2in+2

, . . . , xn+kin+k

)∥∥∥2

H S

⎞⎠

12

≤ [Bp]n

πmultp (T ) ‖λ‖

lmn+1 ···mn+ks

.

Now, since 12 = 1

p + 1s and by taking the supremum over all ‖λ‖l

mn+1 ···mn+ks

≤ 1, the leftmember becomes

⎛⎝

mn+1,...,mn+k∑in+1,...,in+k=1

∥∥∥T(

xn+1in+1

, xn+2in+2

, . . . , xn+kin+k

)∥∥∥p

H S

⎞⎠

1p

≤ [Bp]n

πmultp (T ).

From here, by an homogeneous argument, we obtain

⎛⎝

mn+1,...,mn+k∑in+1,...,in+k=1

∥∥∥T(

xn+1in+1

, xn+2in+2

, . . . , xn+kin+k

)∥∥∥p

H S

⎞⎠

1p

≤ [Bp]n

πmultp (T ) · wp(xn+1

in+1| 1 ≤ in+1 ≤ mn+1) · · · wp

(xn+k

in+k| 1 ≤ in+k ≤ mn+k

),

and (ii) is proved.(ii)⇒ (i) Again from [17, Proposition 5.7] and [23, Theorem 4.2], we have H S(H1, . . . ,

Hn; H) = �multp (H1, . . . , Hn; H) and [Ap]nπmult

p (·) ≤ ‖ · ‖H S . Then from (ii) we deduce

that T : Xn+1 × · · · × Xn+k → �multp (H1, . . . , Hn; H) is multiple p-summing with

respect to the multiple p-summing norm on �multp (H1, . . . , Hn; H). Since, as we have

already remarked, the implication (β) ⇒ (γ ) is always true, it follows that T is multiplep-summing.

As a consequence of the main result, we are able to find and prove some coincidenceresults for the multiple p-summing operators. Establishing such results is a very importantand intensively studied problem. In this direction, we know an inclusion result, obtained byPérez García in [23, Theorem 3.6] and later, a coincidence result which was obtained byPopa in [29, Theorem 6, Theorem 10] and independently by Botelho and Pellegrino in [4,5].In addition, Popa has also shown an inclusion result, see [29]. Thus, it was proved thefollowing result

Theorem 2 (a) If all Banach spaces X1, . . . , Xn have cotype 2 and Y has also cotype 2,then for all 1 ≤ p ≤ 2, we have the coincidence

�multp (X1, . . . , Xn; Y ) = �mult

1 (X1, . . . , Xn; Y ).

(b) If Y has cotype 2, then for any Banach spaces X1, . . . , Xn and all 2 ≤ p < ∞, we havethe reverse inclusion

�multp (X1, . . . , Xn; Y ) ⊂ �mult

2 (X1, . . . , Xn; Y ).

As a consequence of the Theorem 1, we can indicate a situation where the inclusion fromTheorem 2 (b) becomes a coincidence. Further, item (i) in the next Corollary has no linearanalogue, while item (ii) is a completion of Theorem 2 (a).

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Corollary 1 (i) Let X be a Banach space and H1, . . . , Hn, H be Hilbert spaces. Thenfor each 2 ≤ p < ∞, we have the coincidence

�multp (H1, . . . , Hn, X; H) = �mult

2 (H1, . . . , Hn, X; H) .

(ii) Let X be a Banach space of cotype 2 and H1, . . . , Hn, H be Hilbert spaces. Then foreach 1 ≤ p < ∞, we have the coincidence

�multp (H1, . . . , Hn, X; H) = �mult

2 (H1, . . . , Hn, X; H) .

Proof (i) Let 2 ≤ p < ∞. By the Theorem 2 (b), we have to prove only the inclusion�mult

2 (H1, . . . , Hn, X; H) ⊆ �multp (H1, . . . , Hn, X; H).

Let T ∈ �mult2 (H1, . . . , Hn, X; H). Then, by the Theorem 1, we have that T : X →

H S(H1, . . . , Hn; H) is a linear 2-summing operator with respect to the Hilbert–Schmidt norm on H S (H1, . . . , Hn; H). By the well known inclusion result in thelinear case, we deduce that T : X → H S (H1, . . . , Hn; H) is p-summing and thus,again by the Theorem 1, T ∈ �mult

p (H1, . . . , Hn, X; H).

(ii) The coincidence �multp (H1, . . . , Hn, X; H) = �mult

2 (H1, . . . , Hn, X; H) is theconsequence of item (i) and the Theorem 2, item (a).

3 Some examples of summing operators

As an application of Theorem 1 and Corollary 1, we give some concrete examples. Our firstkind of operator is the bilinear extension of the well known multiplication linear operatorsand the second one is a bilinear extension of Hardy’s operator. In order to obtain the neces-sary and sufficient conditions for such a kind of operators to be multiple summing, we needto know when a linear multiplication operator is summing. In the case of linear operators,Garling in [15, Theorem 9] has given an almost complete description of the summing prop-erties for the multiplication operators. For the sake of completeness, we will state and provehere only those situations from Garling’s Theorem which are needed in our paper.

We recall that if a = (an)n∈N, b = (bn)n∈N are two scalar sequences, we write ab =(anbn)n∈N. For a = (an)n∈N a scalar sequence, we denote by Ma the multiplication operatorwhich acts between two sequence spaces and it is defined by Ma(x) = ax . As it is wellknown, if 1 ≤ p, q < ∞ then Ma : lq → l p given by Ma(x) = ax is well defined if andonly if a ∈ l∞ for q ≤ p, or a ∈ lr for p < q , where 1

p = 1q + 1

r .

Proposition 2 (a) Let 1 ≤ s < ∞ and Ma : ls → l2 be the multiplication operator. Then

(i) If 1 ≤ s ≤ 2, 1 ≤ p < ∞, Ma is p-summing if and only if a ∈ ls∗ ;(ii) If 2 < s, 2 ≤ p < ∞, Ma is p-summing if and only if a ∈ l2;

(iii) If 2 < s, 1 < s∗ ≤ p ≤ 2, Ma is p-summing if and only if a ∈ l p;(iv) If 2 < s, 1 ≤ p ≤ s∗ ≤ 2, Ma is p-summing if and only if a ∈ ls∗ .

(b) Let 1 ≤ s ≤ 2 and Ma : l2 → ls be the multiplication operator. If 1 ≤ p < ∞, Ma isp-summing if and only if a ∈ ls .

Proof (a) We prove the followingClaim (1). If 1 ≤ s < ∞ and a ∈ ls∗ , then Ma : ls → l2 is 1 -summing, hence is

p-summing for each 1 ≤ p < ∞.

Indeed, since a ∈ ls∗ , then Ma : ls → l2 has the factorization lsMa→ l1

I↪→ l2 and since,

by Grothendieck’s Theorem, the inclusion I : l1 ↪→ l2 is 1-summing, the claim is proved.

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(i) For 1 ≤ s ≤ 2, ls has cotype 2 and, since l2 has also cotype 2, by [13, Corol-lary 11.16(c)], [28, Corollary 5.17], [32, Corollary 10.18], we have the coincidence�p(ls, l2) = �1(ls, l2) for each 1 ≤ p < ∞. Hence, in this case, it is enough to findthe necessary and sufficient conditions for Ma to be 1-summing.Suppose that Ma is 1-summing. For each b ∈ ls, we have w1(bnen | n ∈ N; ls) =‖Mb : ls∗ → l1‖ and, since Ma is 1-summing, it follows that (‖Ma(bnen)‖)n∈N ∈ l1i.e. ab ∈ l1. Thus we are in the situation: for each b ∈ ls it follows that ab ∈ l1, which,as it is well known, implies a ∈ ls∗ . The converse follows from the Claim (1).

(ii) Since l2 has cotype 2, by [13, Theorem 11.13], [28, Theorem 5.15], [32, Corollary10.18(iii)], we have the coincidence �p(ls, l2) = �2(ls, l2) for each 2 ≤ p < ∞.Hence, in this case, it is enough to find the necessary and sufficient conditions for Ma

to be 2-summing.Suppose that Ma is 2-summing. Since 2 < s < ∞, then s∗ < 2 and w2(en | n ∈N; ls) = ‖I : ls∗ ↪→ l2‖ = 1. Thus, it follows that (‖Ma(en)‖)n∈N ∈ l2 i.e. a ∈ l2.Conversely, suppose a ∈ l2. Then from the fact that w2(en | n ∈ N; l2) = ‖I : l2 → l2‖= 1,

∑∞n=1 ‖anen‖2 = ‖a‖2

2, it follows that Ma = ∑∞n=1 anen ⊗ en is a 2-nuclear

representation of Ma , which means that Ma is 2-nuclear and hence, 2-summing, see[13, Proposition 5.5, Theorem 5.27], [26, Proposition 18.2.2].

(iii) Suppose that Ma is p-summing. Then from s∗ ≤ p we have wp(en | n ∈ N; ls) = ‖I :ls∗ ↪→ l p‖ = 1 and the fact that Ma is p-summing, gives us that (‖Ma(en)‖)n∈N ∈ l p ,i.e. a ∈ l p .Conversely, suppose a ∈ l p . Since 1 < p ≤ 2, we have 2 ≤ p∗ < ∞ and thenwp∗(en | n ∈ N; l2) = ‖I : l2 ↪→ l p∗‖ = 1,

∑∞n=1 ‖anen‖p = ‖a‖p

p . This means thatMa = ∑∞

n=1 anen ⊗ en is a p-nuclear representation of Ma , i.e. Ma is p-nuclear, andhence p-summing, [13, Proposition 5.5, Theorem 5.27], [26, Proposition 18.2.2].

(iv) Suppose that Ma is p-summing. Since p ≤ s∗, by the inclusion theorem, it follows thatMa is s∗-summing. Then from ws∗(en | n ∈ N; ls) = ‖I : ls∗ → ls∗‖ = 1 and the factthat Ma is s∗-summing, we get (‖Ma(en)‖)n∈N ∈ ls∗ , i.e. a ∈ ls∗ .The converse follows from the Claim (1).We note that the assertions (iii) and (iv) can be written under the form: If 1 ≤ p,

s∗ ≤ 2, Ma is p-summing if and only if a ∈ lmax(p,s∗).(b) Since for 1 ≤ s ≤ 2, ls has cotype 2, by [13, Corollary 11.16(c)], [28, Corollary

5.17], [32, Corollary 10.18], we have the coincidence �p(ls, l2) = �1(ls, l2) for each1 ≤ p < ∞, so in this case it is enough to find the necessary and sufficient conditionsfor Ma to be 2-summing.

Suppose that Ma is 2-summing. For each b ∈ ls∗ , Mb : ls → l1 is well defined. Now

we consider the diagram l2Ma→ ls

Mb→ l1I

↪→ l2. By Grothendieck’s Theorem l1I

↪→ l2 is2-summing and since Ma is also 2-summing, then by another Grothendieck’s Theorem, see[13, Theorem 5.31], we have that the composition J ◦ Mb ◦ Ma : l2 → l2 is nuclear. ButJ ◦ Mb ◦ Ma = Mab, thus Mab : l2 → l2 is nuclear. Then, as it is well known and easyto prove, this implies ab ∈ l1. Thus we are in the situation: for each b ∈ ls∗ it follows thatab ∈ l1 which, as it is well known, implies that a ∈ ls .

Conversely, if a ∈ ls , then ws∗(en | n ∈ N; ls) = ‖I : ls∗ → ls∗‖ = 1,∑∞

n=1 ‖anen‖s =‖a‖s

s . This means that Ma = ∑∞n=1 anen ⊗ en is a s-nuclear representation of Ma , i.e. Ma is

s-nuclear, and hence s-summing, thus since s ≤ 2, Ma is 2-summing.

In what follows, based on our main result, Theorem 1, Corollary 1 and Proposition 2,we will present two examples of bilinear multiple p-summing operators.

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First, we consider the bilinear analogue of the multiplication operator, namely if 1 ≤p, q, r < ∞ and a is a sequence of scalars, we define Ba : lr × lq → l p by the formulaBa(x, y) = axy. As it is well known, Ba is well defined if and only if a ∈ l∞ in the case1p ≤ 1

q + 1r , or a ∈ ls in the case 1

p > 1q + 1

r , where 1p = 1

q + 1r + 1

s .

Proposition 3 Let 1 ≤ s < ∞, a ∈ l∞ and let Ba : l2 × ls → l2 be the bilinear multiplica-tion operator.

(i) If 1 ≤ s ≤ 2, 1 ≤ p < ∞, Ba is multiple p-summing if and only if a ∈ ls∗ ;(ii) If 2 < s, 2 ≤ p < ∞, Ba is multiple p-summing if and only if a ∈ l2;

(iii) If 2 < s, 1 < s∗ ≤ p < 2, Ba is multiple p-summing if and only if a ∈ l p;(iv) If 2 < s, 1 < p ≤ s∗ < 2, Ba is multiple p-summing if and only if a ∈ ls∗ .

Proof First we proveClaim (2). If 1 ≤ s ≤ 2, Ba is multiple 2-summing if and only if a ∈ ls∗ ;Claim (3). If 2 < s, Ba is multiple 2 -summing if and only if a ∈ l2.Indeed, by the Theorem 1, Ba is multiple 2-summing if and only if Ba : ls → H S (l2, l2)

is 2-summing with respect to the Hilbert–Schmidt norm on H S(l2, l2). If y ∈ ls , thenBa(y) = May : l2 → l2, thus

∥∥Ba (y)∥∥

H S = ‖ay‖l2 = ‖Ma (y)‖l2 .

From here we deduce that Ba : ls → H S(l2, l2) is 2-summing with respect to the Hilbert–Schmidt norm on H S(l2, l2) if and only if Ma : ls → l2 is 2-summing, which by Proposition2 (a), (i) and (ii) proves the Claim (2) and respectively, the Claim (3).

(i) Since for 1 ≤ s ≤ 2, ls has cotype 2, by Corollary 1 (ii) we have the coincidence�mult

p (l2, ls; l2) = �mult2 (l2, ls; l2) for each 1 ≤ p < ∞ and the statement follows

from Claim (2).(ii) By Corollary 1 (i), we have that for each 2 ≤ p < ∞,�mult

p (l2, ls; l2) = �mult2 (l2, ls;

l2) and then, the statement follows from Claim (3).We observe that (iii) and (iv) do not follow from Theorem 1 and Corollary 1.

(iii) Suppose that Ba is multiple p-summing. From s∗ ≤ p, we have that wp(en | n ∈N; ls) = ‖I : ls∗ ↪→ l p‖ = 1 i.e. η = (en)n∈N ∈ wp(ls). Since Ba is multiplep-summing, by definition, it follows that (Ba)η : l2 → l p(l2) defined by (Ba)η(x) =(Ba(x, en))n∈N is p-summing. For each x ∈ l2, we have Ba(x, en) = an〈x, en〉en andthus

∥∥(Ba)η (x)∥∥

l p(l2)= ‖Ma (x)‖l p

.

Since (Ba)η is p-summing, this equality implies that Ma : l2 → l p is p-summing andthen, since 1 < p ≤ 2, from the Proposition 2 (b) we have that a ∈ l p .Conversely, suppose a ∈ l p . We first prove that Ba : ls → H S(l2, l2), is p-summingwith respect to the Hilbert–Schmidt norm on H S(l2, l2). Indeed, if y ∈ ls , then

∥∥Ba (y)∥∥

H S = ‖ay‖l2 = ‖Ma (y)‖l2 .

From here we deduce that Ba : ls → H S(l2, l2) is p-summing with respect to theHilbert–Schmidt norm on H S(l2, l2) if and only if Ma : ls → l2 is p-summing,which by the hypothesis and the Proposition 2 (a), (iii) holds. By [13, Corollary11.16], [32, Corollary 10.18], or, in this case, by the old result of Pełczynski’s [20],we have H S(l2, l2) = �p(l2, l2). Hence Ba : ls → �p(l2, l2) is p-summing. As wehave already observed the implication (β) ⇒ (γ ) is always true, thus Ba is multiplep-summing.

123


(iv) Suppose that Ba is multiple p-summing. From 1 ≤ p ≤ s∗ < 2, by the inclusiontheorem for multiple summing operators, see [23, Theorem 3.6], it follows that Ba

is multiple s∗-summing. We have ws∗(en | n ∈ N; ls) = ‖I : ls∗ → ls∗‖ = 1 i.e.η = (en)n∈N ∈ ws∗(ls). Then, since Ba is multiple s∗-summing, by definition it fol-lows that (Ba)η : l2 → ls∗(l2) defined by (Ba)η(x) = (Ba(x, en))n∈N is s∗-summing.As in (iii),

∥∥(Ba)η (x)∥∥

ls∗ (l2)= ‖Ma (x)‖ls∗ for each x ∈ l2.

Since (Ba)η is s∗-summing from this equality we deduce that Ma : l2 → ls∗ is s∗-summing.Since 1 < s∗ ≤ 2, by the Proposition 2 (b), we obtain a ∈ ls∗ .

Conversely, suppose a ∈ ls∗ . In this case, we first prove that Ba : ls → H S(l2, l2) is1-summing. Indeed, for each y ∈ ls we have

∥∥Ba (y)∥∥

H S = ‖ay‖l2 = ‖Ma (y)‖l2 ,

from where we deduce that Ba : ls → H S(l2, l2) is 1-summing if and only if Ma : ls → l2is 1-summing, which by the Claim (1) and the fact that a ∈ ls∗ holds. Next, by the inclu-sion theorem in the linear case, we will deduce that Ba : ls → H S(l2, l2) is p-summing.Then by the old result of Pełczynski’s, [20], we have H S(l2, l2) = �p(l2, l2) and henceBa : ls → �p(l2, l2) is p-summing. Since the implication (β) ⇒ (γ ) is always true, we getthat Ba is multiple p-summing. ��

Our next result gives an example of a multiple p-summing operator, p ≥ 1, which is anextension of the well known Hardy operator.

Proposition 4 Let a = (an)n∈N ∈ l∞, 1 ≤ s < ∞ and Ha : l2 × ls → l2,

Ha (x, y) =(

a1 〈x, e1〉〈y, e1〉 + · · · + an 〈x, en〉〈y, en〉n

)

n∈N

.

(i) If 1 ≤ s ≤ 2, 1 ≤ p < ∞, Ha is multiple p- summing if and only if(

an√n

)n∈N

∈ ls∗ ;

(ii) If 2 < s, 2 ≤ p < ∞, Ha is multiple p-summing if and only if(

an√n

)n∈N

∈ l2;

(iii) If 2 < s, 1 < s∗ ≤ p < 2, Ha is multiple p-summing if and only if(

an√n

)n∈N

∈ l p;

(iv) If 2 < s, 1 ≤ p ≤ s∗ < 2, Ha is multiple p-summing if and only if(

an√n

)n∈N

∈ ls∗ .

Proof As we will see, the proof is analogous the one given for Proposition 3.

Let’s consider the sequence b = (bn)n∈N, bn =(∑∞

k=n1k2

) 12. We will further need the

fact that ab ∈ lt if and only if(

an√n

)n∈N

∈ lt . Indeed, as a simple consequence of the

Stolz–Cesaro Theorem for the case[ 0

0

], we have that lim

n→∞b2

n1n

= 1, which is equivalent to

limn→∞ anbn

an1√n

= 1, hence∑∞

n=1(anbn)t converges if and only if∑∞

n=1

(an√

n

)tconverges.

(i) For 1 ≤ s ≤ 2, ls has cotype 2. Then, by the Corollary 1 (ii), we have �multp (l2, ls; l2) =

�mult2 (l2, ls; l2). Thus, it is enough to find the necessary and sufficient conditions for Ha

to be multiple 2-summing. By the Theorem 1, Ha is multiple 2-summing if and onlyif Ha : ls → H S(l2, l2) is 2-summing. Let y ∈ ls . Then Hay(en) = Ha(en, y) =(0, . . . , 0,

an〈y,en〉n ,

an〈y,en〉n+1 , . . .) and

123


∥∥Hay∥∥2

H S =∞∑

n=1

∥∥Hay (en)∥∥2

l2=

∞∑n=1

|an |2 |〈y, en〉|2 b2n = ‖Mab(y)‖2

l2 , (2)

where Mab : ls → l2 is the multiplication operator. Hence, Ha : ls → H S(l2, l2)is 2-summing with respect to the Hilbert–Schmidt norm on H S(l2, l2) if and only if Mab is2-summing, which by the Proposition 2 (a), (i), is equivalent to ab ∈ ls∗ . This condition is

equivalent to(

an√n

)n∈N

∈ ls∗ .

(ii) By the Theorem 1, Ha is multiple p-summing if and only if Ha : ls → H S(l2, l2)is p-summing, which, by using (2), is equivalent to Mab : ls → l2 is p-summing. By the

Proposition 2 (a), (ii), this is equivalent to ab ∈ l2, which is equivalent to(

an√n

)n∈N

∈ l2.

We observe that (iii) and (iv) do not follow from Theorem 1 and Corollary 1.(iii) Suppose that Ha is multiple p-summing. From s∗ ≤ p, we have that wp(en | n ∈

N; ls) = ‖I : ls∗ ↪→ l p‖ = 1 i.e. η = (en)n∈N ∈ wp(ls). Since Ha is multiple p-summing,by definition, it follows that (Ha)η : l2 → l p(l2) defined by (Ha)η(x) = (Ha(x, en))n∈N is

p-summing. For each x ∈ l2, we have Ha(x, en) =(

0, . . . , 0,an〈x,en〉

n ,an〈x,en〉

n+1 , . . .)

∈ l2,

hence

‖Ha (x, en)‖l2 = |anbn 〈x, en〉|and thus

∥∥(Ha)η (x)∥∥

l p(l2)= ‖Mab (x)‖l p

.

Since (Ha)η is p-summing, the previous equality implies that Mab : l2 → l p is p-summing,which since 1 < p ≤ 2, by the Proposition 2 (b), means that ab ∈ l p .

Conversely, suppose ab ∈ l p . In this case, we first prove that Ha : ls → H S(l2, l2)is p-summing. Indeed, for each y∈ ls we have ‖Ha(y)(en)‖l2 =‖Ha(en, y)‖l2 =|anbn〈y, en〉|and

∥∥Ha (y)∥∥

H S = ‖aby‖l2 = ‖Mab (y)‖l2 .

This equality proves that Ha : ls → H S(l2, l2) is p-summing if and only if Mab : ls → l2is p-summing. Since by the hypothesis we have ab ∈ l p , the Proposition 2 (a), (iii) gives usthat Mab : ls → l2 is indeed p-summing. Thus Ha : ls → H S(l2, l2) is p-summing. By [13,Corollary 11.16], [32, Corollary 10.18], or, in this case, by the old result of Pełczynski’s [20],we have H S(l2, l2) = �p(l2, l2) and thus Ha : ls → �p(l2, l2) is p-summing. As we havealready observed, the implication (β) ⇒ (γ ) is always true, thus Ha is multiple p-summing.

The statement follows from the equivalence: ab ∈ l p if and only if(

an√n

)n∈N

∈ l p .

(iv) Suppose that Ha is multiple p-summing. From 1 ≤ p ≤ s∗ < 2, by the inclusiontheorem for multiple summing operators, see [23, Theorem 3.6], it follows that Ha is multiples∗-summing. We have ws∗(en | n ∈ N; ls) = ‖I : ls∗ → ls∗‖ = 1 i.e. η = (en)n∈N ∈ ws∗(ls).Since Ha is multiple s∗-summing, by definition, it follows that (Ha)η : l2 → ls∗(l2) definedby (Ha)η(x) = (Ha(x, en))n∈N is s∗-summing. As in (iii), we have

∥∥(Ha)η (x)∥∥

ls∗ (l2)= ‖Mab (x)‖ls∗ for each x ∈ l2.

Since (Ha)η is s∗-summing, this equality proves that Mab : l2 → ls∗ is s∗-summing. Since1 < s∗ ≤ 2, the Proposition 2 (b) implies that ab ∈ ls∗ .

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Conversely, suppose ab ∈ ls∗ . In this case, we first prove that Ha : ls → H S(l2, l2) is1-summing. Indeed, let y ∈ ls . Then ‖Ha(y)(en)‖l2 = |anbn〈y, en〉| and

∥∥Ha(y)∥∥

H S = ‖aby‖l2 = ‖Mab(y)‖l2 .

Hence Ha : ls → H S(l2, l2) is 1-summing if and only if Mab : ls → l2 is 1-summing, whichholds by invoking Claim (1) and using the fact that ab ∈ ls∗ . Next, by the inclusion theoremin the linear case, we will deduce that Ha : ls → H S(l2, l2) is p-summing. As in (iii), wededuce that Ha : ls → �p(l2, l2) is p-summing, which since the implication (β) ⇒ (γ ) isalways true, gives us that Ha is multiple p-summing.

The statement follows from the fact that ab ∈ ls∗ is equivalent to(

an√n

)n∈N

∈ ls∗ .

Acknowledgments The authors thank the referee for helpful suggestions.

References

1. Acosta, M., García, D., Maestre, M.: A multilinear Lindenstrauss theorem. J. Funct. Anal. 235,122–136 (2006)

2. Bohnenblust, H.F., Hille, E.: On the absolute convergence of Dirichlet series. Ann. Math. 32,600–622 (1931)

3. Bombal, F., Pérez-García, D., Villanueva, I.: Multilinear extensions of Grothendieck’s theorem.Q. J. Math. 55, 441–450 (2004)

4. Botelho, G., Pellegrino, D.: Coincidences for multiple summing mappings, In: Segundo Enama, 27–28,Joao Pessoa, Brazil (2008)

5. Botelho, G., Michels, C., Pellegrino, D.: Complex interpolation and summability properties of multilinearoperators. Revista Matematica Complutense 23, 139–161 (2010)

6. Botelho, G., Pellegrino, D.: When every multilinear mapping is multiple summing. Mathematische Nach-richten 282, 1414–1422 (2009)

7. Çaliskan, E., Pellegrino, D.: On the multilinear generalizations of the concept of absolutely summingoperators. Rocky Mt. J. Math. 37, 1137–1154 (2007)

8. Defant, A., Floret, K.: Tensor norms and operator ideals, North-Holland (1993)9. Defant, A., García, D., Maestre, M., Pérez-García, D.: Bohr’s strip for vector valued Dirichlet series.

Mathematische Annalen 342, 533–555 (2008)10. Defant, A., Pérez-García, D.: A tensor norm preserving unconditionality for Lp-spaces. Trans. Am. Math.

Soc. 360, 3287–3306 (2008)11. Defant, A., Sevilla-Peris, P.: A new multilinear insight on Littlewood’s 4/3-inequality. J. Funct. Anal.

256, 1642–1664 (2009)12. Defant, A., Popa, D., Schwarting, U.: Coordinatewise multiple summing operators in Banach spaces.

J. Funct. Anal. 259(1), 220–242 (2010)13. Diestel, J., Jarchow, H., Tonge, A.: Absolutely summing operators. Cambridge University Press,

Cambridge (1995)14. Dineen, S.: Complex Analysis on Infinite Dimensional Spaces. Springer, London (1999)15. Garling, D.J.H.: Diagonal maps between sequence spaces. Stud. Math. 51, 129–138 (1974)16. Littlewood, J.E.: On bounded bilinear forms in an infinite number of variables. Q. J. (Oxford Ser.) 1, 164–

174 (1930)17. Matos, M.C.: Fully absolutely summing and Hilbert–Schmidt multilinear mappings. Collect.

Math. 54, 111–136 (2003)18. Matos, M.C., Pellegrino, D.: Fully summing mappings between Banach spaces. Stud. Math. 178, 47–

61 (2007)19. Mujica, J.: Complex Analysis in Banach Spaces. North-Holland, Amsterdam (1986)20. Pełczynski, A.: A characterization of Hilbert–Schmidt operators. Stud. Math. 28, 355–360 (1967)21. Pellegrino, D., Souza, M.: Fully summing multilinear and holomorphic mappings into Hilbert

spaces. Mathematische Nachrichten 278, 877–887 (2005)22. Peréz-García, D., Villanueva, I.: Multiple summing operators on Banach spaces. J. Math. Anal. Appl.

285, 86–96 (2003)

123


23. Pérez-García, D.: The inclusion theorem for multiple summing operators. Stud. Math. 165(3),275–290 (2004)

24. Pérez-García, D., Villanueva, I.: Multiple summing operators on C(K) spaces. Ark. for Mat. 42, 153–171(2004)

25. Peréz-García, D.: Comparing different classes of absolutely summing multilinear operators. Archiv derMathematik (Basel) 85, 258–267 (2005)

26. Pietsch, A.: Operator Ideals, North Holland (1980)27. Pietsch, A.: Ideals of multilinear functionals (designs of a theory). In: Proceedings of second International

Conference Operator Alg, Leipzig, Germany, Teubner-Texte zur Mathematik 62 , pp. 185–199 (1983)28. Pisier, G.: Factorization of linear operators and geometry of Banach spaces. CBMS Regional Conf.

Series 60, Amer. Math. Soc. (1986)29. Popa, D.: Reverse inclusions for multiple summing operators. J. Math. Anal. Appl. 350(1), 360–368 (2009)30. Popa, D.: Multilinear variants of Maurey and Pietsch theorems and applications. J. Math. Anal.

Appl. 368(1), 157–168 (2010)31. Ramanujan, M.S., Schock, E.: Operator ideals and spaces of bilinear operators. Linear Multilinear Alge-

bra 18, 307–318 (1985)32. Tomczak-Jagermann, N.: Banach-Mazur distances and finite dimensional operator ideals, Pitman Monogr.

Surveys Pure Appl. Math., vol. 38, Longman Scientific & Technical/Wiley, Harlow/New York (1989)

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Hilbert–Schmidt and multiple summing operators