Positive almost Dunford–Pettis operators and their duality

We study some properties of almost Dunford–Pettis operators and we characterize pairs of Banach lattices for which the adjoint of an almost Dunford–Pettis operator inherits the same property and look at conditions under which an operator is almost Dunford–Pettis whenever its adjoint is.     

The Alternative Dunford–Pettis Property for Subspaces of the Compact Operators

A Banach space X has the alternative Dunford–Pettis property if for every weakly convergent sequences (x_n) → x in X and (x_n*) → 0 in X* with ||x_n|| = ||x||= 1 we have (x_n*(x_n)) → 0. We get a characterization of certain operator spaces having the alternative Dunford–Pettis property. As a consequence of this result, if H is a Hilbert space we show that a closed subspace M of the compact operators on H has the alternative Dunford–Pettis property if, and only if, for any h ∈ H, the evaluation operators from M to H given by S ↦ Sh, S ↦ S^th are DP1 operators, that is, they apply weakly convergent sequences in the unit sphere whose limits are also in the unit sphere into norm convergent sequences. We also prove a characterization of certain closed subalgebras of K(H) having the alternative Dunford-Pettis property by assuming that the multiplication operators are DP1.     

Some characterizations of almost Dunford–Pettis operators and applications

This paper attempts to deal with some characterizations of almost Dunford–Pettis operators from a Banach lattice into a Banach space. It also discusses some of the consequences derived from this study. As an application, we generalize some results of Meyer-Nieberg on the duality between semi-compact operators and order weakly compact operators.     

Convolution factorability of bilinear maps and integral representations

In this paper we consider a special class of continuous bilinear operators acting in a product of Banach algebras of integrable functions with convolution product. In the literature, these bilinear operators are called ‘zero product preserving’, and they may be considered as a generalization of Lamperti operators. We prove a factorization theorem for this class, which establishes that each zero product preserving bilinear operator factors through a subalgebra of absolutely integrable functions. We obtain also compactness and summability properties for these operators under the assumption of some classical properties for the range spaces, as the Dunford–Pettis property or the Schur property and we give integral representations by some concavity properties of operators. Finally, we give some applications for integral transforms, and an integral representation for Hilbert–Schmidt operators.     

On positive almost weak* Dunford–Pettis operators

In this paper, we introduce the class of almost weak* Dunford–Pettis operators and give a characterization of this class of operators. We study its relation with the classes of weak* Dunford–Pettis operators and almost Dunford–Pettis operators, and its relation with the closely related classes of almost limited operators and L-weakly compact operators.     

On the class of weak^{\star } Dunford–Pettis operators

We give a characterization of weak\(^{\star }\) Dunford–Pettis operators and we study its relation with limited completely continuous operators.     

Toeplitz Operators with Special Symbols on Segal–Bargmann Spaces

We study the boundedness of Toeplitz operators on Segal–Bargmann spaces in various contexts. Using Gutzmer’s formula as the main tool we identify symbols for which the Toeplitz operators correspond to Fourier multipliers on the underlying groups. The spaces considered include Fock spaces, Hermite and twisted Bergman spaces and Segal–Bargmann spaces associated to Riemannian symmetric spaces of compact type.     

Orlicz–Morrey Spaces and Fractional Operators

By taking an interest in a natural extension to the small parameters of the trace inequality for Morrey spaces, Orlicz–Morrey spaces are introduced and some inequalities for generalized fractional integral operators on Orlicz–Morrey spaces are established. The local boundedness property of the Orlicz maximal operators is investigated and some Morrey-norm equivalences are also verified. The result obtained here sharpens the one in our earlier papers.     

On Cesàro summability of vector valued multiplier spaces and operator valued series

In this paper, we introduce and study vector valued multiplier spaces with the help of the sequence of continuous linear operators between normed spaces and Cesàro convergence. Also, we obtain a new version of the Orlicz–Pettis Theorem by means of Cesàro summability.     

Towards an L^p Potential Theory for Sub-Markovian Semigroups： Kernels and Capacities

We study in fairly general measure spaces （X,μ） the （non-linear） potential theory of L^p sub-Markovian semigroups which are given by kernels having a density with respect to the underlying measure. In terms of mapping properties of the operators we provide sufficient conditions for the existence （and regularity） of such densities. We give various （dual） representations for several associated capacities and, in the corresponding abstract Bessel potential spaces, we study the role of the truncation property. Examples are discussed in the case of R^n where abstract Bessel potential spaces can be identified with concrete function spaces.     

Moreau–Yosida Regularization of Maximal Monotone Operators of Type (D)

We propose a Moreau–Yosida regularization for maximal monotone operators of type (D), in non-reflexive Banach spaces. It generalizes the classical Moreau–Yosida regularization as well as Brezis–Crandall–Pazy’s extension of this regularization to strictly convex (reflexive) Banach spaces with strictly convex duals. Our main results are obtained by making use of recent results by the authors on convex representations of maximal monotone operators in non-reflexive Banach spaces.     

Summing Boolean Algebras

In this paper we will study some familes and subalgebras F of P (N) that let us characterize the unconditional convergence of series through the weak convergence of subseries ∑i∈A^xi, A ∈ F. As a consequence, we obtain a new version of the Orlicz-Pettis theorem, for Banach spaces. We also study some relationships between algebraic properties of Boolean algebras and topological properties of the corresponding Stone spaces.     

On the class of positive almost weak Dunford–Pettis operators

We introduce and study the class of almost weak Dunford–Pettis operators and we derive the following interesting consequence: other characterizations of the weak Dunford–Pettis property. After that we characterize pairs of Banach lattices for which the adjoint of almost weak Dunford–Pettis operator is almost Dunford–Pettis. Finally, we establish a necessary and sufficient conditions on the pair of Banach lattices E and F which guarantees that if T : E → F is a positive almost weak Dunford–Pettis then T is almost Dunford–Pettis.     

Interpolation operators in Orlicz–Sobolev spaces

We study classical interpolation operators for finite elements, like the Scott–Zhang operator, in the context of Orlicz–Sobolev spaces. Furthermore, we show estimates for these operators with respect to quasi-norms which appear in the study of systems of p-Laplace type.     

Universal mappings for certain classes of operators and polynomials between Banach spaces

A well‐known result of J. Lindenstrauss and A. Pełczyński (1968) gives the existence of a universal non‐weakly compact operator between Banach spaces. We show the existence of universal non‐Rosenthal, non‐limited, and non‐Grothendieck operators. We also prove that there does not exist a universal non‐Dunford–Pettis operator, but there is a universal class of non‐Dunford–Pettis operators. Moreover, we show that, for several classes of polynomials between Banach spaces, including the non‐weakly compact polynomials, there does not exist a universal polynomial.     

Amenability of Banach algebras of compact operators

In this paper we study conditions on a Banach spaceX that ensure that the Banach algebraК(X) of compact operators is amenable. We give a symmetrized approximation property ofX which is proved to be such a condition. This property is satisfied by a wide range of Banach spaces including all the classical spaces. We then investigate which constructions of new Banach spaces from old ones preserve the property of carrying amenable algebras of compact operators. Roughly speaking, dual spaces, predual spaces and certain tensor products do inherit this property and direct sums do not. For direct sums this question is closely related to factorization of linear operators. In the final section we discuss some open questions, in particular, the converse problem of what properties ofX are implied by the amenability ofК(X). BEJ supported by MSRVP at Australian National University; GAW supported by SERC grant GR-F-74332.     

Composition Operators on Hardy–Orlicz Spaces on the Ball

We give embedding theorems for Hardy–Orlicz spaces on the ball and then apply our results to the study of the boundedness and compactness of composition operators in this context. As one of the motivations of this work, we show that there exist some Hardy–Orlicz spaces, different from H ^∞, on which every composition operator is bounded.     

Strong Dunford–Pettis Sets and Spaces of Operators

Bibasic sequences of Singer are used to show that ℓ₁ embeds complementably in the Banach space X if and only if X^* contains a non-relatively compact strong Dunford–Pettis set. Spaces of operators and strongly additive vector measures are also discussed. An erratum to this article is available at .     

On contraction properties of Markov kernels

 We study Lipschitz contraction properties of general Markov kernels seen as operators on spaces of probability measures equipped with entropy-like ``distances'. Universal quantitative bounds on the associated ergodic constants are deduced from Dobrushin's ergodic coefficient. Strong contraction properties in Orlicz spaces for relative densities are proved under more restrictive mixing assumptions. We also describe contraction bounds in the entropy sense around arbitrary probability measures by introducing a suitable Dirichlet form and the corresponding modified logarithmic Sobolev constants. The interest in these bounds is illustrated on the example of inhomogeneous Gaussian chains. In particular, the existence of an invariant measure is not required in general. Received: 31 October 2000 / Revised version: 21 February 2003 / Published online: 12 May 2003 L. Miclo also thanks the hospitality and support of the Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brasil, where part of this work was done. Mathematics Subject Classification (2000): 60J05, 60J22, 37A30, 37A25, 39A11, 39A12, 46E39, 28A33, 47D07 Key words or phrases: Lipschitz contraction – Generalized relative entropy – Markov kernel – Dobrushin's ergodic coefficient – Orlicz norm – Dirichlet form – Spectral gap – Modified logarithmic Sobolev inequality – Inhomogeneous Gaussian chains – Loose of memory property     

The <Emphasis Type="Italic">p</Emphasis>-Gelfand–Phillips property in spaces of operators and Dunford–Pettis like sets

The p-Gelfand–Phillips property (1 \({\leq}\) p < ∞) is studied in spaces of operators. Dunford–Pettis type like sets are studied in Banach spaces. We discuss Banach spaces X with the property that every p-convergent operator T:X \({\rightarrow}\) Y is weakly compact, for every Banach space Y.