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

Ioana Ghenciu¹

¹ Department of Mathematics University of Wisconsin--River Falls River Falls, WI 54022-5001, U.S.A.

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 63 (2015) no. 3, pp. 261-274

Résumé

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 $A \in \Sigma$.

Détail

DOI : 10.4064/ba7997-1-2016

Keywords: suppose banach spaces compact hausdorff space sigma sigma algebra borel subsets banach space continuous nobreakdash valued functions supremum norm x strongly bounded operator representing measure sigma strongly bounded operator hat its extension limited only its extension hat limited * completely continuous resp unconditionally converging only hat * completely continuous resp unconditionally converging prove dispersed compact hausdorff space strongly bounded operator limited resp weakly precompact has completely continuous adjoint has unconditionally converging adjoint whenever limited resp weakly precompact has completely continuous adjoint has unconditionally converging adjoint each sigma