Vector lattices of weakly compact operators on Banach lattices

A result of Aliprantis and Burkinshaw shows that weakly compact operators from an AL-space into a KB-space have a weakly compact modulus. Groenewegen characterised the largest class of range spaces for which this remains true whenever the domain is an AL-space and Schmidt proved a dual result. Both of these authors used vector-valued integration in their proofs. We give elementary proofs of both results and also characterise the largest class of domains for which the conclusion remains true whenever the range space is a KB-space. We conclude by studying the order structure of spaces of weakly compact operators between Banach lattices to prove results analogous to earlier results of one of the authors for spaces of compact operators.

     

Topological characterisation of weakly compact operators

Let X be a Banach space. Then there is a locally convex topology for X, the “Right topology,” such that a linear map T, from X into a Banach space Y, is weakly compact, precisely when T is a continuous map from X, equipped with the “Right” topology, into Y equipped with the norm topology. When T is only sequentially continuous with respect to the Right topology, it is said to be pseudo weakly compact. This notion is related to Pelczynski's Property (V).     

Yosida-Hewitt type decompositions for weakly compact operators and operator-valued measures

We derive Yosida-Hewitt type decompositions for weakly compact operators from Köthe-Bochner function spaces to Banach spaces. As an application, we obtain a Yosida-Hewitt type decomposition for strongly bounded operator-valued measures.     

Compact and weakly compact composition operators from the Bloch space into Möbius invariant spaces

We obtain exhaustive results and treat in a unified way the question of boundedness, compactness, and weak compactness of composition operators from the Bloch space into any space from a large family of conformally invariant spaces that includes the classical spaces like BMOA  , _Qα$Q_{\alpha }$, and analytic Besov spaces B^p$B^p$. In particular, by combining techniques from both complex and functional analysis, we prove that in this setting weak compactness is equivalent to compactness. For the operators into the corresponding “small” spaces we also characterize the boundedness and show that it is equivalent to compactness.     

Weakly compact composition operators on vector-valued BMOA

Weak compactness of the analytic composition operator f?f○φ is studied on BMOA(X), the space of X-valued analytic functions of bounded mean oscillation, and its subspace VMOA(X), where X is a complex Banach space. It is shown that the composition operator is weakly compact on BMOA(X) if X is reflexive and the corresponding composition operator is compact on the scalar-valued BMOA. A concrete example is given which shows that BMOA(X) differs from the weak vector-valued BMOA for infinite dimensional Banach spaces X.     

An image problem for compact operators

Let be a separable Banach space and a sequence of closed subspaces of satisfying for all . We first prove the existence of a dense-range and injective compact operator such that each is a dense subset of , solving a problem of Yahaghi (2004). Our second main result concerns isomorphic and dense-range injective compact mappings between dense sets of linearly independent vectors, extending a result of Grivaux (2003).

     

Factoring weakly compact operators and the inhomogeneous Cauchy problem

By using the technique of factoring weakly compact operators through reflexive Banach spaces we prove that a class of ordinary differential equations with Lipschitz continuous perturbations has a strong solution when the problem is governed by a closed linear operator generating a strongly continuous semigroup of compact operators.

     

Order-weakly compact operators from vector-valued function spaces to Banach spaces

Let be an ideal of over a  -finite measure space , and let stand for the order dual of . For a real Banach space let be a subspace of the space of -equivalence classes of strongly -measurable functions and consisting of all those for which the scalar function belongs to . For a real Banach space a linear operator is said to be order-weakly compact whenever for each the set is relatively weakly compact in . In this paper we examine order-weakly compact operators . We give a characterization of an order-weakly compact operator in terms of the continuity of the conjugate operator of with respect to some weak topologies. It is shown that if is an order continuous Banach function space, is a Banach space containing no isomorphic copy of and is a weakly sequentially complete Banach space, then every continuous linear operator is order-weakly compact. Moreover, it is proved that if is a Banach function space, then for every Banach space any continuous linear operator is order-weakly compact iff the norm is order continuous and is reflexive. In particular, for every Banach space any continuous linear operator is order-weakly compact iff is reflexive.

     

Operators with connected spectrum + compact operators = strongly irreducible operators

Herrero’s conjecture that each operator with connected spectrum acting on complex, separable Hilbert space can be written as the sum of a strongly irreducible operator and a compact operator is proved. Jiang, C. L., Power, S., Wang, Z. Y., Biquasitriangular + small compact = strongly irreducible,J. London Math., to be published.     

Centripetal operators and Koszmider spaces

We study properties of Koszmider spaces and introduce a related notion of weakly Koszmider spaces. We show that if the space K is weakly Koszmider and C(K) is isomorphic to C(L) then L is also weakly Koszmider, but the analogous result does not hold for Koszmider spaces. We also show that a connected Koszmider space is strongly rigid.     

Characterizations of weakly compact operators on

Let be a locally compact Hausdorff space and let , is continuous and vanishes at infinity} be provided with the supremum norm. Let and be the -rings generated by the compact subsets and by the compact subsets of , respectively. The members of are called -Borel sets of since they are precisely the -bounded Borel sets of . The members of are called the Baire sets of . denotes the dual of . Let be a quasicomplete locally convex Hausdorff space. Suppose is a continuous linear operator. Using the Baire and -Borel characterizations of weakly compact sets in as given in a previous paper of the author's and combining the integration technique of Bartle, Dunford and Schwartz, we obtain 35 characterizations for the operator to be weakly compact, several of which are new. The independent results on the regularity and on the regular Borel extendability of -additive -valued Baire measures are deduced as an immediate consequence of these characterizations. Some other applications are also included.

     

The structure of compact disjointness preserving operators on continuous functions

Let T be a compact disjointness preserving linear operator from C₀(X) into C₀(Y), where X and Y are locally compact Hausdorff spaces. We show that T can be represented as a norm convergent countable sum of disjoint rank one operators. More precisely, T = Σ_n δ ?h_n for a (possibly finite) sequence {x_n }_n of distinct points in X and a norm null sequence {h_n }_n of mutually disjoint functions in C₀(Y). Moreover, we develop a graph theoretic method to describe the spectrum of such an operator (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Factorizing operators on Banach function spaces through spaces of multiplication operators

In order to extend the theory of optimal domains for continuous operators on a Banach function space X(μ) over a finite measure μ, we consider operators T satisfying other type of inequalities than the one given by the continuity which occur in several well-known factorization theorems (for instance, Pisier Factorization Theorem through Lorentz spaces, pth-power factorable operators …). We prove that such a T factorizes through a space of multiplication operators which can be understood in a certain sense as the optimal domain for T. Our extended optimal domain technique does not need necessarily the equivalence between μ and the measure defined by the operator T and, by using δ-rings, μ is allowed to be infinite. Classical and new examples and applications of our results are also given, including some new results on the Hardy operator and a factorization theorem through Hilbert spaces.     

Two classes of operators with irreducibility and the small and compact perturbations of them

This paper gives the concepts of finite dimensional irreducible operators((FDI) operators)and infinite dimensional irreducible operators((IDI) operators). Discusses the relationships of(FDI)operators,(IDI) operators and strongly irreducible operators((SI) operators) and illustrates some properties of the three classes of operators. Some sufficient conditions for the finite-dimensional irreducibility of operators which have the forms of upper triangular operator matrices are given. This paper proves that every operator with a singleton spectrum is a small compact perturbation of an(FDI) operator on separable Banach spaces and shows that every bounded linear operator T can be approximated by operators in(Σ FDI)(X) with respect to the strong-operator topology and every compact operator K can be approximated by operators in(Σ FDI)(X) with respect to the norm topology on a Banach space X with a Schauder basis, where(ΣFDI)(X) := {T∈B(X) : T=Σki=1Ti, Ti ∈(FDI), k ∈ N}.     

A note on weakly compact subsets in the projective tensor product of Banach spaces

Let X and Y be two Banach spaces. In this short note we show that every weakly compact subset in the projective tensor product of X and Y can be written as the intersection of finite unions of sets of the form , where K_X and K_Y are weakly compacts subsets of X and Y, respectively. If either X or Y has the Dunford–Pettis property, then any intersection of sets that are finite unions of sets of the form , where K_X and K_Y are weakly compact sets in X and Y, respectively, is weakly compact.     

On algebras of polynomially compact operators

In this paper, we find sufficient and necessary conditions for a triangularizable closed algebra of polynomially compact operators to be commutative modulo the radical. We also prove that an algebraic algebra of operators of a bounded degree on a Banach space is triangularizable under some mild additional conditions. As a special case we obtain a result stating that every algebraic algebra of operators of bounded degree is triangularizable whenever its commutators are nilpotent operators.     

Weakly compact operators into sequence spaces: A counterexample

In a recent paper Gutiérrez and Villanueva have used, without giving a detailed proof, an analogue of a well-known result of Ryan characterizing the weakly compact operators from a Banach space into the space of null sequences in a Banach space . In this note a counterexample is given showing that in the statement of Gutiérrez and Villanueva an additional condition is needed.

     

Strong pseudo-contractions perturbed by compact operators in Banach spaces

Let be a (real) Banach space, let be an open subset of , and let denote the collection of all nonempty bounded and closed subsets of . Suppose is continuous from into with respect to the Hausdorff metric and strongly pseudo-contractive, while is compact from into . Then has a fixed point if it satisfies the classical Leray-Schauder condition on the boundary of .