Strongly bounded operators on with the strict topology

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• 作者：Marian Nowak ^{M.Nowak@wmie.uz.zgora.pl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Spaces of vector-valued continuous functions ; Strict topologies ; Operator measures ; Strongly bounded operators ; Unconditionally converging operators ; Weakly compact operators ; Weakly precompact operators ; Completely continuous operators ; Weakly completely continuous operators
• 刊名：Indagationes Mathematicae
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：27
• 期：4
• 页码：972-984
• 全文大小：264 K

文摘

Let $X$ be a completely regular Hausdorff space, and $E$ and $F$ be Banach spaces. Let $C_{rc}(X,E)$ be a space of all continuous functions $f:X\to E$ such that $f\left(X\right)$ is a relatively compact set in $E$, equipped with the strict topology ${\beta }_{\sigma }$. We study $({\beta }_{\sigma },{\Vert \cdot \Vert }_F)$-continuous strongly bounded operators $T:C_{rc}(X,E)\to F$. In particular, we establish the relationships between $({\beta }_{\sigma },{\Vert \cdot \Vert }_F)$-continuous strongly bounded operators and weakly compact (resp. weakly precompact; unconditionally converging; completely continuous; weakly completely continuous) operators $T:C_{rc}(X,E)\to F$. In particular, it is shown that if $E$ is a Schur space, then the space $(C_{rc}(X,E),{\beta }_{\sigma })$ has the strict Dunford–Pettis property.