Continuous linear operators onC(K, X) and pointwise weakly precompact subsets ofC(K, X)

Abstract

Let K be a compact Hausdorif space, X a Banach space and C(K, X) the Banach space of all continuous functions : KX equipped with the supremum norm. A subset H of C(K, X) is pointwise weakly precompact if, for each t in K, the set Ht) = {(t):H} is weakly precompact. In this note we study the images of a bounded pointwise weakly precompact subset H of C(K, X) under several classes of linear operators on C(K, X).

Complemented Subspaces of Linear Bounded Operators

“…(iii) By results in [6,31], C(K, X) has property (V ). Hence, by part (i), every operator T : C(K, X) → Y is unconditionally converging, and thus weakly compact.…”

Strongly Bounded Representing Measures and Convergence Theorems

“…(iii) If T : C(K, X) → Y is an unconditionally converging operator, then T is a Dieudonné operator, since X has property (u) [32]. By Corollary 3, T is weakly compact.…”

Properties (V) and (w V) on C(Ω X)

“…We begin by investigating the structure of V subsets of duals of arbitrary Banach spaces. We first consider the Properties (V) and (wV) on C(Q.,X) 471 following characterization of weakly conditionally compact sets given by A. Ulger in [16].…”

“…LEMMA [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. A subset K of M(Q.,X*) is weakly conditionally compact if and only if K F is weakly conditionally compact for each separable subspace F of X.…”