Strictly singular and strictly cosingular operators on spaces of continuous functions

Abstract

In this paper we will be concerned with studying operatorsT:C(K, X) →Ydefined on Banach spaces of continuous functions. We will be particularly interested in studying the classes of strictly singular and strictly cosingular operators. In the process, we obtain answers to certain questions recently raised by Bombal and Porras in [5]. Specifically, we study Banach spaceXandYfor which an operatorT:C(K, X) →Ywith representing measuremis strictly singular (strictly cosingular) whenevermis strongly bounded andm(A) is strictly singular (strictly cosingular) for each Borel subsetAofK. Along the way we establish several results dealing with non-compact operators on continuous function spaces, and we consolidate numerous results concerning extension theorems for operators defined on these same spaces. Also, we join Saab and Saab [25] in demonstrating that if l1does not embed inX* then the adjointT* of a strongly bounded map must be weakly precompact, thereby presenting an alternative solution to a question raised in [2].