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On Some Classes of Operators on $C(K,X)$

Volume 63 / 2015

Ioana Ghenciu Bulletin Polish Acad. Sci. Math. 63 (2015), 261-274 MSC: Primary 46B20; Secondary 28B05. DOI: 10.4064/ba7997-1-2016 Published online: 7 January 2016

Abstract

Suppose $X$ and $Y$ are Banach spaces, $K$ is a compact Hausdorff space, $\Sigma$ is the $\sigma$-algebra of Borel subsets of $K$, $C(K,X)$ is the Banach space of all continuous $X$-valued functions (with the supremum norm), and $T:C(K,X)\to Y$ is a strongly bounded operator with representing measure $m:\Sigma \to L(X,Y)$.

We show that if $T$ is a strongly bounded operator and $\hat{T}: B(K, X) \to Y$ is its extension, then $T$ is limited if and only if its extension $\hat{T}$ is limited, and that $T^*$ is completely continuous (resp. unconditionally converging) if and only if $\hat{T}^*$ is completely continuous (resp. unconditionally converging).

We prove that if $K$ is a dispersed compact Hausdorff space and $T$ is a strongly bounded operator, then $T$ is limited (resp. weakly precompact, has a completely continuous adjoint, has an unconditionally converging adjoint) whenever $m(A):X\to Y$ is limited (resp. weakly precompact, has a completely continuous adjoint, has an unconditionally converging adjoint) for each $A \in \Sigma$.

Authors

  • Ioana GhenciuDepartment of Mathematics
    University of Wisconsin--River Falls
    River Falls, WI 54022-5001, U.S.A.
    e-mail

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