P-summing Operators Research Articles

Absolutely simple p-summing operators and applications

Absolutely simple p-summing operators and applications

Lipschitz 𝑝-summing multilinear operators correspond to Lipschitz 𝑝-summing operators

We give conditions that ensure that an operator satisfying a Piestch domination in a given setting also satisfies a Piestch domination in a different setting. From this we derive that a bounded multilinear operator T T is Lipschitz p p -summing if and only if the mapping f T ( x 1 ⊗ ⋯ ⊗ x n ) ≔ T ( x 1 , … , x n ) f_T(x_1\otimes \cdots \otimes x_n)≔T(x_1,\ldots , x_n) is Lipschitz p p -summing. The results are based on the projective tensor norm. An example with the Hilbert tensor norm is provided to show that the statement may not hold when a reasonable cross-norm other than the projective tensor norm is considered.

Further results on strictly Lipschitz summing operators

Abstract The aim of this paper is to give some new characterizations of strictly Lipschitz p-summing operators. These operators have been introduced in order to improve the Lipschitz p-summing operators. Therefore, we adapt this definition for constructing other classes of Lipschitz mappings which are called strictly Lipschitz p-nuclear and strictly Lipschitz (p, r, s)-summing operators. Some interesting properties and factorization results are obtained for these new classes.

Positively p-nuclear operators, positively p-integral operators and approximation properties

In the present paper, we introduce and investigate a new class of positively p-nuclear operators that are positive analogues of right p-nuclear operators. One of our main results establishes an identification of the dual space of positively p-nuclear operators with the class of positive p-majorizing operators that is a dual notion of positive p-summing operators. As applications, we prove the duality relationships between latticially p-nuclear operators introduced by O. I. Zhukova and positively p-nuclear operators. We also introduce a new concept of positively p-integral operators via positively p-nuclear operators and prove that the inclusion map from $$L_{p^{*}}(\mu )$$ to $$L_{1}(\mu )$$ ( $$\mu $$ finite) is positively p-integral. New characterizations of latticially p-integral operators and positively p-integral operators are presented and used to prove that an operator is latticially p-integral (resp. positively p-integral) precisely when its second adjoint is. Finally, we describe the space of positively p-integral operators as the dual of the $$\Vert \cdot \Vert _{\Upsilon _{p}}$$ -closure of the subspace of finite rank operators in the space of positive p-majorizing operators. Approximation properties, even positive approximation properties, are needed in establishing main identifications.

Some properties of almost summing operators

In this paper we extend the scope of three important results in the linear theory of absolutely summing operators. The first one was obtained by Bu and Kranz in [4] and it asserts that a continuous linear operator between Banach spaces takes almost unconditionally summable sequences into Cohen strongly q-summable sequences for any q≥2, whenever its adjoint is p-summing for some p≥1. The second of them states that p-summing operators with hilbertian domain are Cohen strongly q-summing operators (1<p,q<∞), this result is due to Bu [3]. The third one is due to Kwapień [8] and it characterizes spaces isomorphic to a Hilbert space using 2-summing operators. We will show that these results are maintained replacing the hypothesis of the operator to be p-summing by almost summing. We will also give an example of an almost summing operator that fails to be p-summing for every 1≤p<∞.

Absolutely (q, 1)-summing operators acting in C(K)-spaces and the weighted Orlicz property for Banach spaces

We provide a new separation-based proof of the domination theorem for (q, 1)-summing operators. This result gives the celebrated factorization theorem of Pisier for (q, 1)-summing operators acting in C(K)-spaces. As far as we know, none of the known versions of the proof uses the separation argument presented here, which is essentially the same that proves Pietsch Domination Theorem for p-summing operators. Based on this proof, we propose an equivalent formulation of the main summability properties for operators, which allows to consider a broad class of summability properties in Banach spaces. As a consequence, we are able to show new versions of the Dvoretzky–Rogers Theorem involving other notions of summability, and analyze some weighted extensions of the q-Orlicz property.

Positive p-summing operators and disjoint p-summing operators

In the present paper, we introduce a new concept of positive p-majorizing operators as a dual notion of positive p-summing operators and generalize the concept of majorizing operators introduced by Schaefer (Isr J Math 13:400–415, 1972). We introduce the concept of positive (p, q)-dominated operators and prove a positive version of the famous Kwapien’s factorization theorem for (p, q)-dominated operators via positive p-majorizing operators. We also introduce the notion of disjoint p-summing operators which is a new larger class of operators than positive p-summing operators and use it to characterize the Radon–Nikodým property. Finally, we investigate the maximal properties of these four classes of operators and prove that they are maximal in corresponding sense.

The Lipschitz injective hull of Lipschitz operator ideals and applications

We introduce and study the Lipschitz injective hull of Lipschitz operator ideals defined between metric spaces. We show some properties and apply the results to the ideal of Lipschitz p-nuclear operators, obtaining the ideal of Lipschitz quasi p-nuclear operators. Also, we introduce in a natural way the ideal of Lipschitz Pietsch p-integral operators and show that its Lipschitz injective hull coincide with the ideal of Lipschitz p-summing operators defined by Farmer and Johnson. Finally, we consider both ideals as Lipschitz operator ideals between a metric space and a Banach space, showing that these ideals are not of composition type. Their maximal hull and minimal kernel are also studied.

Stochastic Integration with Respect to Cylindrical L\xe9vy Processes by p-Summing Operators

We introduce a stochastic integral with respect to cylindrical Lévy processes with finite p-th weak moment for pin [1,2]. The space of integrands consists of p-summing operators between Banach spaces of martingale type p. We apply the developed integration theory to establish the existence of a solution for a stochastic evolution equation driven by a cylindrical Lévy process.

Operators with the Maurey–Pietsch multiple splitting property

Let n be a natural number, such that . As a simultaneous extension of the p-summing and multiple p-summing operators we introduce the class of all n-linear operators which are -summing, . If , are such that we introduce the class of the operators with the Maurey–Pietsch -splitting property as the multilinear analogue of the class of -mixing linear operators and, as particular classes, , . We introduce two new classes of multilinear operators both being the multilinear analogues of the class of r-nuclear operators, namely, the class of r-nuclear operators with respect to the -norm, and the class and prove that . By concrete examples, we show that, contrary to the linear case, when , in the multilinear case the inclusions , are not true, which is in contrast to a well-known result shown by us in another paper that.

On p-Dunford integrable functions with values in Banach spaces

Let (Ω,Σ,μ) be a complete probability space, X a Banach space and 1≤p<∞. In this paper we discuss several aspects of p-Dunford integrable functions f:Ω→X. Special attention is paid to the compactness of the Dunford operator of f. We also study the p-Bochner integrability of the composition u∘f:Ω→Y, where u is a p-summing operator from X to another Banach space Y. Finally, we also provide some tests of p-Dunford integrability by using w⁎-thick subsets of X⁎.

Characterizations of new Cohen summing bilinear operators

We prove two characterizations of new Cohen summing bilinear operators. The first one is: Let X, Y and Z be Banach spaces, 1 < p < ∞, V : X × Y → Z a bounded linear operator and n ≥ 2 a natural number. Then V is new Cohen p-summing if and only if for all Banach spaces X1, … , Xn and all p-summing operators U : X1 × · · · × Xn → X, the operator V ◦ (U, IY) : X1 × · · · × Xn × Y → Z is -summing. The second result is: Let H be a Hilbert space,, Y, Z Banach spaces and V : H × Y → Z a bounded bilinear operator and 1 < p < ∞. Then V is new Cohen p-summing if and only if for all Banach spaces E and all p-summing operators U : E → H, the operator V ◦ (U, IY) is (p, p*)-dominated.

Domination spaces and factorization of linear and multilinear summing operators

It is well known that not every summability property for multilinear operators leads to a factorization theorem. In this paper we undertake a detailed study of factorization schemes for summing linear and nonlinear operators. Our aim is to integrate under the same theory a wide family of classes of mappings for which a Pietsch type factorization theorem holds. Our construction includes the cases of absolutely p-summing linear operators, (p, σ)-absolutely continuous linear operators, factorable strongly p-summing multilinear operators, (p1, . , pn)-dominated multilinear operators and dominated (p1, . , pn; σ)-continuous multilinear operators.

A note on the concept of factorable strongly p-summing operators

It is known that an n-homogeneous polynomial is factorable strongly p-summing if and only if its linearization is absolutely p-summing. We prove that a similar result holds for n-linear operators and, as a consequence, we present simplified proofs of recent results from Pellegrino et al. (Rev R Acad Cienc Exactas Fis Nat Ser A Mat RACSAM 110(1):285–302, 2016). Using this connection and the well-known results of the classical linear theory of absolutely p-summing operators we prove other properties for this class of multilinear operators.

The splitting property for $$(\,p,S)$$ ( p , S ) -summing operators

We introduce the unifying concept of (p, S)-summing operator which include the well-known concept of p-summing operator as well the concept of summing operator introduced by Kislyakov and Blasco-Signes. We prove a Pietsch’s domination theorem and a splitting lemma from which we derive a Pietsch’s composition type result for this new class. The new feature of our splitting result is that the proof is entirely different than those known in the literature.

Some properties of Lipschitz strongly p-summing operators

We consider the space of molecules endowed with the transposed version of the Chevet–Saphar norm and we identify its dual space with the space of Lipschitz strongly p-summing operators. We also extend some old results to the category of Lipschitz mappings and we give a factorization result of Lipschitz (p,r,s)-summing operators.

On the Hadamard Sequence Spaces h p (X) and the p-Summing Operators

We consider the p-HH-norms and the Hadamard sequence space hp(X), where X is a normed space. We discuss the dual space of hp(X). We consider a similar definition to that of the p-summing operator, where we equip the Banach space with the p-HH-norms instead of the lp norms, which we refer to as the p-HH-summing operator and discuss its properties. We discuss a set-valued p-summing operator, and connect this to the p-summing and the p-HH-summing operators.

Cohen and multiple Cohen strongly summing multilinear operators

Considering the successful theory of multiple summing multilinear operators as a prototype, we introduce the classes of multiple Cohen strongly p-summing multilinear operators and polynomials. The adequacy of these classes under the viewpoint of the theory of multilinear and polynomial ideals and holomorphy types is discussed in detail.

Interpolative construction and factorization of operators

Banach operator ideals generated by interpolative construction applied to p-summing operators are studied. These ideals are described in terms of factorization through abstract interpolation Lorentz spaces. Relationships between Banach ideals determined by Orlicz sequence spaces are shown and a variant of the Pisier factorization theorem for (p,1)-summing operators from C(K)-spaces is proved. Applications to Schatten classes are given. It is also shown that certain known results on (q,p)-concave operators from Banach lattices can be lifted to a class of generalized concave operators.

The invariant subspace problem for absolutely p-summing operators in Krein spaces

Let 1≤p<∞, and let T be a bounded linear operator defined on a Krein space K. We prove the existence of a non-positive subspace L − of K invariant under T with the assumption that T is absolutely p-summing with some further conditions imposed on it.MSC:47B50, 46C20, 47B10.