Positive p-Summing Operators on L p -Spaces

Carpathian Mathematical Publications, 2020

In 2003, Dimant V. has defined and studied the interesting class of strongly $p$-summing multilinear operators. In this paper, we introduce and study a new class of operators between two Banach lattices, where we extend the previous notion to the positive framework, and prove, among other results, the domination, inclusion and composition theorems. As consequences, we investigate some connections between our class and other classes of operators, such as duality and linearization.

Demonstratio Mathematica, 2000

A summable sequence (a n) in a Banach space X is called Zp-canonical, 1 < p < oo, if a n = a n ve n , n = 1,2 where (a n) € lp, v : l p-• X is a continuous linear operator and (e n) is the natural basis of l p. We are showing that a summable sequence (a n) in X is Zp-canonical iff the operator u : cq-• X, with ue n = a n , n = 1,2,... is psumming. It follows that in a given Banach space X any summable sequence is Zp-canonical iff any continuous linear operator from co to X is p-summing. The last assertion implies the following statement obtained previously in [Kva]: in a given Banach space X any summable sequence is Zp-canonical for certain p, 2 < p < oo iff X does not contain Z£o's uniformly. For the spaces with a given cotype p we are obtaining the more precise results showing, in particular, that in cotype 2 spaces any summable sequence is Z2-canonical, while in l p , with 2 < p < oo, not any summable sequence is Zp-canonical.

Collectanea Mathematica, 2015

We prove new summability properties for multilinear operators on ℓ p spaces. An important tool for this task is a better understanding of the interplay between almost summing and absolutely summing multilinear operators.

Abstract and Applied Analysis, 2001

We give necessary and sufficient conditions for an operator on the spaceC (T,X)to be(r,p)-absolutely summing. Also we prove that the injective tensor product of an integral operator and an(r,p)-absolutely summing operator is an(r,p)-absolutely summing operator.

Journal of Mathematical Analysis and Applications, 2005

Let X and Y be Banach spaces and u be a continuous linear operator from X to Y . We prove that if u * , the adjoint operator of u, is p-summing for some p 1, then for any q 2, u takes (almost) unconditionally summable sequences in X into members of q⊗ Y , the projective tensor product of q and Y .

2014

We prove new summability properties for multilinear operators on ℓ_p spaces. An important tool for this task is a better understanding of the interplay between almost summing and absolutely summing multilinear operators.

arXiv (Cornell University), 2014

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.

Positivity, 2018

The aim of this work is to give and study the notion of Cohen positive p-summing multilinear operators. We prove a natural analog of the Pietsch domination theorem for these classes and characterize their conjugates. As an application, we generalize a result due to Bu and Shi (J. Math. Anal. Appl. 401:174-181, 2013), and we compare this class with the class of multiple p-convex m-linear operators.

International Journal of Mathematics and Mathematical Sciences, 1986

The representing vector measure for T need not be of bounded variation. Further, if G is of bounded variation and F doesn't have the RadonoNikodym property, then T need not be a kernel integral operator.

Illinois Journal of Mathematics

It is well known in Banach space theory that for a finite dimensional space E there exists a constant c E , such that for all sequences (x k) k ⊂ E one has k x k ≤ c E sup ε k ±1 k ε k x k. Moreover, if E is of dimension n the constant c E ranges between √ n and n. This implies that absolute convergence and unconditional convergence only coincide in finite dimensional spaces. We will characterize Banach spaces X, where the constant c E ∼ √ n for all finite dimensional subspaces. More generally, we prove that an estimate c E ≤ cn 1− 1 q holds for all n ∈ IN and all n-dimensional subspaces E of X if and only if the eigenvalues of every operator factoring through ℓ ∞ decrease of order k − 1 q if and only if X is of weak cotype q, introduced by Pisier and Mascioni. We emphasize that in contrast to Talagrand's equivalence theorem on cotype q and absolutely (q, 1)-summing spaces this extends to the case q = 2. If q > 2 and one of the conditions above is satisfied one has k x k q 1 q ≤ C 1+l (1 + log 2) (l) ((1 + log 2 n) 1 q) IE k ε k x k for all n, l ∈ IN and (x k) k ⊂ E, E a n dimensional subspace of X. In the case q = 2 the same holds if we replace the expected value by the supremum.

Indagationes Mathematicae, 1978

We follow the notation of [7,10]. In particular, if E[?] is a locally convex space (in short l. c. s. ), ?(E; E,); 1(E; E,) and 炉(E; E,) will denote, respecti-vely, the weak, Mackey and strong topology corresponding to the dual pair hE; E,i. If U is a neighbourhood of 0 in E[?], EU denotes the Banach space associated to U and 谩U : E ! EU denotes the corresponding quotient map.

2005

We prove additivity of the minimal conditional entropy associated with a quantum channel Phi, represented by a completely positive (CP), trace-preserving map, when the infimum of S(gamma_{12}) - S(gamma_1) is restricted to states of the form gamma_{12} = (I \ot Phi)(| psi >< psi |). We show that this follows from multiplicativity of the completely bounded norm of Phi considered as a map from L_1 -> L_p for L_p spaces defined by the Schatten p-norm on matrices; we also give an independent proof based on entropy inequalities. Several related multiplicativity results are discussed and proved. In particular, we show that both the usual L_1 -> L_p norm of a CP map and the corresponding completely bounded norm are achieved for positive semi-definite matrices. Physical interpretations are considered, and a new proof of strong subadditivity is presented.

Illinois Journal of Mathematics, 1989

1993

It is well known in Banach space theory that for a finite dimensional space E there exists a constant c E , such that for all sequences (x k) k ⊂ E one has k x k ≤ c E sup ε k ±1 k ε k x k. Moreover, if E is of dimension n the constant c E ranges between √ n and n. This implies that absolute convergence and unconditional convergence only coincide in finite dimensional spaces. We will characterize Banach spaces X, where the constant c E ∼ √ n for all finite dimensional subspaces. More generally, we prove that an estimate c E ≤ cn 1− 1 q holds for all n ∈ IN and all n-dimensional subspaces E of X if and only if the eigenvalues of every operator factoring through ℓ ∞ decrease of order k − 1 q if and only if X is of weak cotype q, introduced by Pisier and Mascioni. We emphasize that in contrast to Talagrand's equivalence theorem on cotype q and absolutely (q, 1)-summing spaces this extends to the case q = 2. If q > 2 and one of the conditions above is satisfied one has k x k q 1 q ≤ C 1+l (1 + log 2) (l) ((1 + log 2 n) 1 q) IE k ε k x k for all n, l ∈ IN and (x k) k ⊂ E, E a n dimensional subspace of X. In the case q = 2 the same holds if we replace the expected value by the supremum.