Weighted composition operators on the predual and dual Banach spaces of Beurling algebras on Z

Let vv be a weight sequence on ZZ and let ��,����,�� be complex-valued functions on ZZ such that ��(Z)���Z��(Z)���Z. In this paper we study the boundedness, compactness and weak compactness of weighted composition operators C��,��C��,�� on predual Banach spaces c0(Z,1/v)c0(Z,1/v) and dual Banach spaces ������(Z,1/v)������(Z,1/v) of Beurling algebras ���1(Z,v)���1(Z,v).

On Gruenhage spaces, separating ��-isolated families, and their relatives

We prove that, given a topological space X, the following conditions are equivalent. (��) X is a Gruenhage space. (��) X has a countable cover by sets of small local diameter (property SLD) by F���G sets. (��) X has a separating ��-isolated family M���F���G. (��) X has a one-to-one continuous map into a metric space which has a ��-isolated base of F���G sets. Besides, we provide an example which shows Fragmentability ��� property SLD ��� the space to be Gruenhage.