Banach lattices of L-weakly and M-weakly compact operators

E Bayram,A W Wickstead

doi:10.1007/s00013-017-1026-3

Abstract

We give conditions for the linear span of the positive L-weakly compact (resp. M-weakly compact) operators to be a Banach lattice under the regular norm, for that Banach lattice to have an order continuous norm, to be an AL-space or an AM-space.

Highlights

We use [6] as our standard reference about Banach lattices and operators on them but, for the convenience of the reader, let us recall the definitions of the operators that this work involves
There is a considerable literature concerning the relationship between L-weakly compact operators and M-weakly compact operators, on the one hand, with weakly compact operators and a myriad of
Prompted by known results about the spaces of all regular operators, and the linear span of the positive compact or weakly compact operators, natural questions to ask are: When does a domination property hold? When are our spaces Banach lattices? When is the norm in a Banach lattice of operators nice? E.g., when is it order continuous, a KB-norm, an AL-norm, or an AM-norm? We present at least partial answers to all of these questions in this note, apart from answering when they are KB-norms

Summary

Introduction

We use [6] as our standard reference about Banach lattices and operators on them but, for the convenience of the reader, let us recall the definitions of the operators that this work involves. An operator T : E → Y , where E is a Banach lattice and Y a Banach space, is called M-weakly compact if whenever (xn) is a norm bounded disjoint sequence in E, we have T xn → 0. 212], that an L-weakly compact subset of a Banach lattice F is contained in F a, the maximal ideal in F on which the norm is order continuous.

Results

Conclusion