P-summing Operators Research Articles

A general Extrapolation Theorem for absolutely summing operators

In this note, we prove a general version of the Extrapolation Theorem for absolutely summing operators, extending the classical theorem due to B. Maurey [‘Théorèmes de factorisation pour les opérateurs à valeurs dans les espaces Lp’, Soc. Math. France, Asterisque 11, Paris, 1974]. Our result also contains the recent Extrapolation Theorem for Lipschitz p-summing operators as a particular case and also provides new extrapolation-type theorems.

Duality for Lipschitz p-summing operators

Building upon the ideas of R. Arens and J. Eells (1956) [1] we introduce the concept of spaces of Banach-space-valued molecules, whose duals can be naturally identified with spaces of operators between a metric space and a Banach space. On these spaces we define analogues of the tensor norms of Chevet (1969) [3] and Saphar (1970) [14], whose duals are spaces of Lipschitz p-summing operators. In particular, we identify the dual of the space of Lipschitz p-summing operators from a finite metric space to a Banach space — answering a question of J. Farmer and W.B. Johnson (2009) [6] — and use it to give a new characterization of the non-linear concept of Lipschitz p-summing operator between metric spaces in terms of linear operators between certain Banach spaces. More generally, we define analogues of the norms of J.T. Lapresté (1976) [11], whose duals are analogues of A. Pietschʼs ( p , r , s ) -summing operators (A. Pietsch, 1980 [12]). As a special case, we get a Lipschitz version of ( q , p ) -dominated operators.

Operator ranges and spaceability

Recent contributions on spaceability have overlooked the applicability of results on operator range subspaces of Banach spaces or Fréchet spaces. Here we consider general results on spaceability of the complement of an operator range, some of which we extend to the complement of a union of countable chains of operator ranges. Applications we give include spaceability of the non-absolutely convergent power series in the disk algebra and of the non-absolutely p-summing operators between certain pairs of Banach spaces. Another application is to ascent and descent of countably generated sets of continuous linear operators, where we establish some closed range properties of sets with finite ascent and descent.

Remarks on Lipschitz 𝑝-summing operators

In this paper, a nonlinear version of the Extrapolation Theorem is proved and, as a corollary, a nonlinear version of Grothendieck’s Theorem is presented. Finally, we prove that if T : X → H T:X\to H is Lipschitz with X X being a pointed metric space and T ( 0 ) = 0 T(0)=0 such that T # | H ∗ T^\#|_{H^*} is q q -summing ( 1 ≤ q > ∞ ) (1\le q>\infty ) , then T T is Lipschitz 1-summing.

Domination problems for strictly singular operators and other related classes

We survey recent results on domination properties of strictly singular operators and related operator ideals, as well as Banach–Saks operators, Narrow operators and p-summing operators.

Hilbert–Schmidt and multiple summing operators

We characterize the multiple p-summing operators for p ≥ 2, in the situation where the range and some of the factors of the cartesian product are Hilbert spaces. As a consequence, we complete some recent results obtained in this area. Thus, we prove a coincidence 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 to be multiple p-summing, p ≥ 1.

Lipschitz $p$-summing operators

The notion of Lipschitz p-summing operator is introduced. A nonlinear Pietsch factorization theorem is proved for such operators, and it is shown that a Lipschitz p-summing operator that is linear is a p-summing operator in the usual sense.

Preduals and Nuclear Operators Associated with Bounded, p-Convex, p-Concave and Positive p-Summing Operators

AbstractWe use Krivine's form of the Grothendieck inequality to renorm the space of bounded linear maps acting between Banach lattices. We construct preduals and describe the nuclear operators associated with these preduals for this renormed space of bounded operators as well as for the spaces of p-convex, p-concave and positive p-summing operators acting between Banach lattices and Banach spaces. The nuclear operators obtained are described in terms of factorizations through classical Banach spaces via positive operators.

Compactness properties of bounded subsets of spaces of vector measure integrable functions and factorization of operators

Using compactness properties of bounded subsets of spaces of vector measure integrable functions and a representation theorem for q-convex Banach lattices, we prove a domination theorem for operators between Banach lattices. We generalize in this way several classical factorization results for operators between these spaces, as psumming operators.

Comparing different classes of absolutely summing multilinear operators

We compare the classical definitions of p-summing multilinear operators with the class of multiple p-summing operators, by showing which linear properties are properly generalized for each one of these classes.

Multilinear extensions of Grothendieck's theorem

We introduce a new class of multilinear p-summing operators, which we call multiplep-summing. Using them, we can prove several multilinear generalizations of Grothendieck's ‘fundamental theorem of the metric theory of tensor products’. As an application we prove a vector-valued Littlewood's inequality.

Vector measure range duality and factorizations of (D, p)-summing operators from Banach function spaces

We characterize the relationship between the space L1(λ’) and the dual L’1(λ) of the space L1(λ), where (λ, λ’) is a dual pair of vector measures with associated spaces of integrable functions L1(λ) and L1(λ’) respectively. Since the result is rather restrictive, we introduce the notion of range duality in order to obtain factorizations of operators from Banach function spaces that are dominated by the integration map associated to the vector measure λ. We obtain in this way a generalization of the Grothendieck-Pietsch Theorem for p-summing operators.

Operator Ideals, Orthonormal Systems and Lacunary Sets

Given an orthonormal system B in some L2(u) we consider the operator ideals IIB and TB of B-summing and B-type operators and some related ideals. We characterize by certain weak compactness properties when IIB is equal to the operator ideal II2 of 2-summing operators. In lose that B consists of characters of a compact abelian group we characterize when IIB coincides with the operator ideal IIγ of Gauss-summing operators and when TB coincides with the operator ideal IIp of type-2 operators. Moreover, we give a necessary and sufficient condition for Fig to contain the operator ideal IIp of p-summing operators (2 < p < ∞) and for TB to contain the operator ideal Γp of p - factorable operators.

The Grothendieck-Pietsch domination principle for nonlinear summing integral operators

We transform the concept of p-summing operators, 1≤ p < ∞, to the more general setting of nonlinear Banach space operators. For 1-summing operators on B(Σ,X)-spaces having weak integral representations we generalize the Grothendieck-Pietsch domination pri

On the tensor product of weakly compact operators

The identity operator on an infinite dimensional Hilbert space shows that neither the projective nor the injective tensor product of two weakly compact operators will be weakly compact in general; see Holub, [H1, Remark, p. 10]. There are, however, conditions ensuring this. Along the studies of Diestel and Faires, [DF], and Saksman and Tylli, [ST], we will add some more. Proposition 1, due to Saksman and Tylli, deals with the tensor product of a compact and a weakly compact operator. Answering a question of theirs, we consider in Proposition 2 the tensor product of two weakly compact operators on spaces with the Dunford-Pettis property. Having treated the case of absolutely p-summing operators in Proposition 3, we show in Proposition 4 that the injective tensor product of two weakly compact operators on Pisier spaces is weakly compact. Our results concern only the projective and the injective tensor product. For the elementary properties of weakly compact operators to be used here we refer to [DS, VI. 4, p. 482-5].

Antisymmetric tensor products of absolutely p-summing operators

We consider antisymmetric tensor products of absolutely p-summing operators. In connection with this second moments of determinants of random matrices appear. These second moments are closely related to approximation properties of the absolutely 2-summing operators and can be used to characterize some classes of infinite-dimensional Banach spaces. Finite-dimensional results are also obtained by this approach.

P-Summing operators on injective tensor products of spaces

SynopsisLet X, Y and Z be Banach spaces, and let Πp (Y, Z) (1 ≦ p &lt; ∞) denote the space of p-summing operators from Y to Z. We show that, if X is a ℒ∞-space, then a bounded linear operator is 1-summing if and only if a naturally associated operator T#: X → Πl (Y, Z) is 1-summing. This result need not be true if X is not a ℒ∞-space. For p &gt; 1, several examples are given with X = C[0, 1] to show that T# can be p-summing without T being p-summing. Indeed, there is an operator T on whose associated operator T# is 2-summing, but for all N ∈ N, there exists an N-dimensional subspace U of such that T restricted to U is equivalent to the identity operator on . Finally, we show that there is a compact Hausdorff space K and a bounded linear operator for which T#: C(K) → Π1 (l1, l2) is not 2-summing.

Absolutely p-summing operators and Banach spaces containing all $l^{n}_{p}$ uniformly complemented

Absolutely p-summing operators and Banach spaces containing all $l^{n}_{p}$ uniformly complemented

Positive 𝑝-summing operators on 𝐿_{𝑝}-spaces

It is shown that for any Banach space B B every positive p p -summing operator from L p ′ ( μ ) {L^{p’}}(\mu ) in B B , 1 / p + 1 / p ′ = 1 1/p + 1/p’ = 1 , is also cone absolutely summing. We also prove here that a necessary and sufficient condition that B B has the Radon-NikodṔm property is that every positive p p -summing operator T : L p ′ ( μ ) → B T:{L^{p’}}(\mu ) \to B is representable by a function f f in L p ( μ , B ) {L^p}(\mu ,B) .

Order p-summing operators, p-stable distributions and characterizations of LP-spaces

Order p-summing operators, p-stable distributions and characterizations of LP-spaces