Continuous Linear Operators on Spaces of Continuous Functions

Throughout this paper X denotes a bicompact space, B(X) denotes the Banach space of all bounded real-valued functions on X, C(X) denotes the closed linear subspace of B(X) of continuous real-valued functions on X, B(N) is denoted by m (where N is the set of positive integers). The space h(X) is the Banach space of all absolutely summable real-valued functions on X and k denotes k(N). The sequence Banach spaces c0 and lp (1 <p< °o) are respectively the spaces of all real sequences converging to 0 and all pth absolutely summable sequences. The space Li is the Banach space of real-valued Lebesgue integrable functions on [0, l]. If E is a Banach space, then E*, E** denote the first and second duals of E respectively. The weak topology on E is signified by a(E, E*) and the weak* topology on E* is signified by a(E*, E). If E and F are Banach spaces and T is a continuous linear operator (hereafter called an operator) from E to F, then T* denotes the dual operator from F* to E*. If P is onto F, then F is said to be a continuous linear image (c.l.i.) of E. If P maps a neighborhood of 0 in £ into a norm (weak) conditionally compact set in F, then T is said to be compact (weakly compact) .UT maps weak Cauchy sequences in E into norm convergent sequences in F, then T is said to be completely continuous. If the restriction of T to each infinite dimensional linear subspace of E is not a topological linear isomorphism (hereafter called an isomorphism), then P is said to be strictly singular. The purpose of this paper is to study the classes of operators defined above when their domain is a c.l.i. of C(X). In particular, necessary and sufficient conditions are given to make all four of the classes coincide and for them to coincide in pairs. It is shown in (3), (4), (5), (6), (7) and (8) of Theorem 4 that any pair of the classes of operators coincide for domains all c.l.i. of C(X) if and only if X is dispersed. It is well known that if the domain of the operators is C(X), then the completely continuous and the weakly compact operators coincide [4]. This result does not hold, in general, for a c.l.i. of C(X) as is shown in this paper. Recently [l5] it has been shown that if the