Strictly singular and strictly cosingular operators on spaces of continuous functions

Abstract

In this paper we will be concerned with studying operators T : C ( K, X ) → Y defined 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 space X and Y for which an operator T : C ( K, X ) → Y with representing measure m is strictly singular (strictly cosingular) whenever m is strongly bounded and m ( A ) is strictly singular (strictly cosingular) for each Borel subset A of K . 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 l 1 does not embed in X * then the adjoint T * of a strongly bounded map must be weakly precompact, thereby presenting an alternative solution to a question raised in [ 2 ].