Abstract
Let X be a completely regular Hausdorff space and C_b(X) be the space of all bounded continuous functions on X, equipped with the strict topology beta . We study some important classes of (beta ,Vert cdot Vert _E)-continuous linear operators from C_b(X) to a Banach space (E,Vert cdot Vert _E): beta -absolutely summing operators, compact operators and beta -nuclear operators. We characterize compact operators and beta -nuclear operators in terms of their representing measures. It is shown that dominated operators and beta -absolutely summing operators T:C_b(X)rightarrow E coincide and if, in particular, E has the Radon–Nikodym property, then beta -absolutely summing operators and beta -nuclear operators coincide. We generalize the classical theorems of Pietsch, Tong and Uhl concerning the relationships between absolutely summing, dominated, nuclear and compact operators on the Banach space C(X), where X is a compact Hausdorff space.
Highlights
Introduction and preliminariesThe Riesz representation theorem plays a crucial role in the study of operators on the Banach space C(X) of continuous functions on a compact Hausdorff space X
The aim of this paper is to extend these classical results to the setting, where X is a completely regular Hausdorff k-space
It is shown that dominated operators and -absolutely summing operators T ∶ Cb(X) → E coincide and if, in particular, E has the Radon–Nikodym property, -absolutely summing and -nuclear operators T ∶ Cb(X) → E coincide
Summary
The Riesz representation theorem plays a crucial role in the study of operators on the Banach space C(X) of continuous functions on a compact Hausdorff space X. Summing operators between Banach spaces have been the object of several studies B(Bo)) denote the Banach space of all bounded continuous Note that can be characterized as the finest locally convex Hausdorff topology on Cb(X) that coincides with the compact-open topology c on u -bounded sets (see [41, Theorem 2.4]). Let M(X) denote the Banach space of all scalar Radon measures, equipped with the total variation norm ‖ ‖ ∶= (X).
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