Strictly singular and strictly cosingular operators on C(K,E).

Abstract

Let E, F be Banach spaces and K a compact Hausdorff space, L(E, F) the class of bounded linear operators from E into F and C(K,E) the Banach space of continuous E-valued functions defined on K normed by the supremum norm. Every bounded linear operator T:C(K,E) ! F has a representing measure m, i.e., a finitely additive measure defined on the _- field B0(K) of Borel subsets of K with values in L(E, F__) such that Tf = R f dm for each f 2 C(K,E). In an earlier related work certain classes of operators in C(K,E), in particular the weakly compact operators, were studied in terms of their representing measures [Bombal and P. Cembranos, Math. Proc. Cambridge Philos. Soc. 97 (1985), no. 1, 137–146; MR0764502 (86b:47051)]. In this paper the strictly singular and strictly cosingular operators are investigated. It is shown that T is strictly singular if and only if its extension T to B(B0(K),E) is strictly singular, and that, provided the semivariation of m is continuous at ? (e.g. if T is weakly compact), T is strictly cosingular if and only if T is strictly cosingular. The compact dispersed spaces are seen to be those for which “natural” conditions on m are sufficient to ensure that T is strictly singular or strictly cosingular.For such K it is shown that C(K,E) contains a complemented copy of lp (1 _ p < 1) if and only if E does