An absolute continuity for positive operators on Banach lattices

W Feldman,C Piston,Calvin E Piston

doi:10.1155/s016117129100008x

Abstract

For positive operators on a Banach lattice, absolute contnuity conditions are considered. An operator absolutely continuous with respect toTis compared to sums of compositions ofTtogether with orthomorphisms or in special cases projections. Consequences For compact operators on functions spacesC(X)are considered.

Highlights

For positive operators S and T between real Banach lattices several types of "absolute continuity" have been defined
We consider an absolute continuity which will be applicable to spaces which are not necessarily Dedekind complete
Several approximations of an operator absolutely continuous with respect to T are provided in terms of sums of operators of the form Qi0ToHi where Qi and H are orthomorphisms

Summary

INTRODUCTION

For positive operators S and T between real Banach lattices several types of "absolute continuity" have been defined. We say that S is -absolutely continuous with respect to T if for each positive element f in E, we have that SF is in the closure of the order ideal generated by TF. An element e of a Banach lattice E is a quasi-interior point if the order ideal generated by e is dense in E. If for each positive decreasing sequence of functions {fn in E and for each y in the representation space Y, the convergence of Tfn(y) to 0 implies the convergence of Sfn(y) to 0 S is s-absolutely continous with respect to T. If f(O) 0 SF is not in the closure of the order ideal generated by TF and S is not -absolutely continuous with respect to T. Given 6 > 0 and f in C(X), there is an hy in C(X) such that lgy hyll < Elllfll. (see [5], Thm 25.10)

Thus we have

Thus for every g

Thus we have n