%0 Journal Article
%T Properties and in Projective Tensor Products
%A Ioana Ghenciu
%J Journal of Operators
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/405635
%X We give sufficient conditions for a subset of to be relatively weakly compact. A Banach space has property (resp., ) if every -subset of is relatively weakly compact (resp., weakly precompact). We prove that the projective tensor product has property (resp., ), when has property (resp., ), has property , and . 1. Introduction Throughout this paper, , and £¿will denote real Banach spaces. An operator will be a continuous and linear function. The set of all continuous linear transformations from £¿to £¿will be denoted by and the compact operators will be denoted by . In this paper, we study weak precompactness and relative weak compactness in spaces of compact operators. Our results are organized as follows. First, we give sufficient conditions for subsets of to be weakly precompact and relatively weakly compact. These results are used to study whether the projective tensor product has properties and , when and have the respective property. Finally, we prove that, in some cases, if has property , then . 2. Definitions and Notations Our notation and terminology are standard. We denote the canonical unit vector basis of by and the canonical unit vector basis of by . The unit ball of will be denoted by unless otherwise specified, and will denote the continuous linear dual of . The set of all continuous linear transformations from to will be denoted by and the compact and weakly compact operators will be denoted by , respectively . The operator is completely continuous (or Dunford-Pettis) if maps weakly Cauchy sequences to norm convergent sequences. A subset of is said to be weakly precompact provided that every bounded sequence from has a weakly Cauchy subsequence. The Banach space has the Dunford-Pettis property ( ) if every weakly compact operator is completely continuous. A series in is said to be weakly unconditionally convergent (wuc) if, for every , the series is convergent. An operator is unconditionally converging if it maps weakly unconditionally convergent series to unconditionally convergent ones. A bounded subset of is called a -subset of provided that for each wuc series in . Pe£¿czy¨½ski introduced property in his fundamental paper [1]. The Banach space has property if every -subset of is relatively weakly compact. The following results were also established in [1]: reflexive Banach spaces and spaces have property ; the Banach space has property if and only if every unconditionally converging operator from to any Banach space is weakly compact; every quotient space of a Banach space with property has property ; if has property , then is weakly
%U http://www.hindawi.com/journals/joper/2014/405635/