Linear operators and vector measures

Abstract

ABSTRACT. Compact and weakly compact operators on function spaces are studied. Those operators are characterized by properties of finitely additive set functions whose existence is guaranteed by Riesz representation theorems.

Weak compactness of certain sets of measures

“…These results are extended to the case of C(X, E) by many authors ( [4], [1], [17]). In this paper we give necessary and sufficient conditions for an equicontinuous subset of F to be σ(F , F )-compact (i.e.…”

Strongly Bounded Representing Measures and Convergence Theorems

“…(ii) Every weakly compact operator T : C(K, X) → Y is strongly bounded [14]. Then T is completely continuous, and thus C(K, X) has the DPP.…”

“…It follows that if T : C(K, X) → Y has a weakly precompact adjoint, then T is strongly bounded (since it is unconditionally converging) and T * is unconditionally converging (by Theorem 12). [14] andT maps B( , X) into Y (as noted in the Introduction).…”

“…By Corollary 6 of [6], T * x is weakly precompact. Hence T x is an unconditionally converging operator on a C(K)-space, and every unconditionally converging operator on a C(K)-space is weakly compact [14,27] Proof. IfT * is weakly precompact, thenT is unconditionally converging and weakly precompact [6].…”

Norm Attaining Operators and Norming Functionals

“…Further, if f/(E) denotes the uniform closure of the simple functions denned over E, and p corresponds to the weakly compact operator T : f/(E) -► X in the natural manner (cf. Dinculeanu [7], Diestel and Uhl [6, Chapter VI] or Brooks and Lewis [4]), then it is immediate that p attains its diameter iff there is an element x* £ Bx-such that ||T*|| = ||7"(x*)|| (T* is norm attaining [10]). The remainder of the introduction provides some background as well as the setting for the specific versions of this problem that we shall address.…”