Operators on separable L1-predual spaces

We give an extension of the classical Bartle–Dunford–Schwartz theorem for weakly compact operators on a $C(K)$ space, to weakly compact operators on a separable $L^1$-predual space. Using this we show that for operators on these spaces, the set of weakly compact operators that attain their norm is dense in the space of weakly compact operators. For operators from the space of affine continuous functions on a metrizable Choquet simplex with values in an uniformly convex space, we show that the operator theoretic version of the Bishop–Phelps–Bollobás property is valid. This gives an extension of some recent work of Kim and Lee.     

Lipschitz compact operators

We introduce the notion of Lipschitz compact (weakly compact, finite-rank, approximable) operators from a pointed metric space X into a Banach space E. We prove that every strongly Lipschitz p-nuclear operator is Lipschitz compact and every strongly Lipschitz p-integral operator is Lipschitz weakly compact. A theory of Lipschitz compact (weakly compact, finite-rank) operators which closely parallels the theory for linear operators is developed. In terms of the Lipschitz transpose map of a Lipschitz operator, we state Lipschitz versions of Schauder type theorems on the (weak) compactness of the adjoint of a (weakly) compact linear operator.     

Quadratic stochastic operators on Banach lattices

We study the convergence of iterates of quadratic stochastic operators that are mean monotonic. They are defined on the convex set of probability measures concentrated on a weakly compact order interval \(S = [0, f]\) of a fixed Banach lattice F. We study their regularity and identify the limits of trajectories either as the “infimum” or “supremum” of the support of initial distributions.     

Essential norms of weighted composition operators and Aleksandrov measures

We derive exact formulas for the essential and weak essential norms of weighted analytic composition operators acting on certain function spaces in the unit disc, extending and improving earlier results due to Sarason and to Kriete and Moorhouse. Differences of composition operators are also considered. The formulas involve the Aleksandrov measures associated to the symbol of the operator. The results are based on a variant of a general method due to Weis of constructing best compact and weakly compact approximants for linear operators on L¹ spaces.     

Weakly compact operators on non-complete normed spaces

We prove that weakly compact operators on a non-reflexive normed space cannot be bijective. We also show that, in the above result, bijectivity cannot be relaxed to surjectivity. Finally, we study the behaviour of surjective weakly compact operators on a non-reflexive normed space, when they are perturbed by small scalar multiples of the identity, and derive from this study the recent result of Spurný [A note on compact operators on normed linear spaces, Expo. Math. 25 (2007) 261–263] that compact operators on an infinite-dimensional normed space cannot be surjective.     

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.     

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.

     

On the weak compactness of b-weakly compact operators

We characterize Banach lattices on which the class of b-weakly compact operators coincides with that of weakly compact operators.     

The b-Weak Compactness of Order Weakly Compact Operators

We characterize Banach lattices on which the class of order weakly compact operators coincides with that of b-weakly compact operators.     

Composition Operators and Vector-valued BMOA

Analytic composition operators are studied on X-valued versions of BMOA, the space of analytic functions on the unit disk that have bounded mean oscillation on the unit circle, where X is a complex Banach space. It is shown that if X is reflexive and C _φ is compact on BMOA, then C _φ is weakly compact on the X-valued space BMOA _C (X) defined in terms of Carleson measures. A related function-theoretic characterization is given of the compact composition operators on BMOA.     

L-weakly and M-weakly compact operators and the centre

We extend known results concerning the centre of spaces of regular (resp. weakly compact or compact) operators between two Banach lattices to the setting of L-weakly compact and M-weakly compact operators. We also show that the L-weakly compact, M-weakly compact, and compact operators lying in the centre of a Banach lattice coincide.     

The Leray-Schauder condition for 1-set weakly-contractive and (ws)-compact operators

The main purpose of this article is to prove a collection of new fixed point theorems for (ws)-compact and so-called 1-set weakly contractive operators under Leray-Schauder boundary condition. We also introduce the concept of semi-closed operator at the origin and obtain a series of new fixed point theorems for such class of operators. As consequences, we get new fixed point existence for (ws)-compact (in particular nonexpansive) self mappings unbounded closed convex subset of Banach spaces. The main condition in our results is formulated in terms of axiomatic measures of weak noncompactness. Later on, we give an application to generalized Hammerstein type integral equations.     

Vector measures and strict topologies

Let X be a completely regular Hausdorff space and Cb(X) be the space of all real-valued bounded continuous functions on X, endowed with the strict topology βσ. We study topological properties of continuous and weakly compact operators from Cb(X) to a locally convex Hausdorff space in terms of their representing vector measures. In particular, Alexandrov representation type theorems are derived. Moreover, a Yosida-Hewitt type decomposition for weakly compact operators on Cb(X) is given.     

Some Properties of the Class of b-Weakly Compact Operators

We establish several characterizations of b-weakly compact operators one of them is similar to that established by Meyer-Nieberg for order weakly compact operators.     

Ideals of homogeneous polynomials and weakly compact approximation property in Banach spaces

We show that a Banach space E has the weakly compact approximation property if and only if each continuous Banach-valued polynomial on E can be uniformly approximated on compact sets by homogeneous polynomials which are members of the ideal of homogeneous polynomials generated by weakly compact linear operators. An analogous result is established also for the compact approximation property.     

单位球上Hardy空间之间的加权复合算子

We consider the weighted composition operators between Hardy spaces on the unit ball, and obtain some sufficient and necessary conditions of bounded or compact weighted composition operators. We also prove that the operator from H1 to H1 is compact if and only if it is weakly compact. Meanwhile, we get the analogue on the Bergman spaces.     

On weak supercyclicity II

This paper considers weak supercyclicity for bounded linear operators on a normed space. On the one hand, weak supercyclicity is investigated for classes of Hilbertspace operators: (i) self-adjoint operators are not weakly supercyclic, (ii) diagonalizable operators are not weakly l-sequentially supercyclic, and (iii) weak l-sequential supercyclicity is preserved between a unitary operator and its adjoint. On the other hand, weak supercyclicity is investigated for classes of normed-space operators: (iv) the point spectrum of the normed-space adjoint of a power bounded supercyclic operator is either empty or is a singleton in the open unit disk, (v) weak l-sequential supercyclicity coincides with supercyclicity for compact operators, and (vi) every compact weakly l-sequentially supercyclic operator is quasinilpotent.     

Some Results on b-AM-Compact Operators

We characterize Banach lattices for which the class of order weakly compact operators (resp. b-weakly compact operators) between Banach lattices coincide with that of b-AM-compact operators.     

Weakly Noncompact Fixed Point Results of the Schauder and the Krasnosel’skii Type

Invoking the technique of measures of weak noncompactness in Banach spaces, we establish some generalized fixed point results of the Schauder and Krasnoselskii type in the context that the involved operators are not weakly compact. It is shown that the obtained results encompass and extend a number of previously known generalizations of the quoted theorems. Finally, an application is presented to illustrate the results.     

Fixed points, eigenvalues and surjectivity for (ws)-compact operators on unbounded convex sets

The paper studies the existence of fixed points for some nonlinear (ws)-compact, weakly condensing and strictly quasibounded operators defined on an unbounded closed convex subset of a Banach space. Applications of the newly developed fixed point theorems are also discussed for proving the existence of positive eigenvalues and surjectivity of quasibounded operators in similar situations. The main condition in our results is formulated in terms of axiomatic measures of weak noncompactness.