ON SPACES OF WEAK* TO WEAK CONTINUOUS COMPACT OPERATORS

Title & Authors
ON SPACES OF WEAK* TO WEAK CONTINUOUS COMPACT OPERATORS
Kim, Ju Myung;

Abstract
This paper is concerned with the space $\small{\mathcal{K}_{w^*}(X^*,Y)}$ of $\small{weak^*}$ to weak continuous compact operators from the dual space $\small{X^*}$ of a Banach space X to a Banach space Y. We show that if $\small{X^*}$ or $\small{Y^*}$ has the Radon-Nikod$\small{\acute{y}}$m property, $\small{\mathcal{C}}$ is a convex subset of $\small{\mathcal{K}_{w^*}(X^*,Y)}$ with $\small{0{\in}\mathcal{C}}$ and T is a bounded linear operator from $\small{X^*}$ into Y, then $\small{T{\in}\bar{\mathcal{C}}^{{\tau}_{\mathcal{c}}}}$ if and only if $\small{T{\in}\bar{\{S{\in}\mathcal{C}:{\parallel}S{\parallel}{\leq}{\parallel}T{\parallel}\}}^{{\tau}_{\mathcal{c}}}}$, where $\small{{\tau}_{\mathcal{c}}}$ is the topology of uniform convergence on each compact subset of X, moreover, if $\small{T{\in}\mathcal{K}_{w^*}(X^*, Y)}$, here $\small{\mathcal{C}}$ need not to contain 0, then $\small{T{\in}\bar{\mathcal{C}}^{{\tau}_{\mathcal{c}}}}$ if and only if $\small{T{\in}\bar{\mathcal{C}}}$ in the topology of the operator norm. Some properties of $\small{\mathcal{K}_{w^*}(X^*,Y)}$ are presented.
Keywords
$\small{weak^*}$ to weak continuous compact operator;dual of operator space;the topology of compact convergence;approximation properties;
Language
English
Cited by
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2.
Unconditional almost squareness and applications to spaces of Lipschitz functions, Journal of Mathematical Analysis and Applications, 2017, 451, 1, 117
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