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Keywords:
space of totally measurable functions; dominated operator; absolutely summing operator; operator-valued measure
space of totally measurable functions; dominated operator; absolutely summing operator; operator-valued measure
Summary:
Let $\Sigma $ be a $\sigma $-algebra of subsets of a set $\Omega $, and $X$ and $Y$ be Banach spaces. Let $B(\Sigma ,X)$ stand for the Banach space of all $X$-valued totally measurable functions on $\Omega $, equipped with the supremum norm. We study absolutely summing operators ${T\colon B(\Sigma ,X)\rightarrow Y}$. We characterize absolutely summing operators $T\colon B(\Sigma ,X)\rightarrow Y$ in terms of their representing operator-valued measures. It is shown that the classes of dominated operators and absolutely summing operators $T\colon B(\Sigma ,X)\rightarrow Y$ coincide if and only if every bounded linear operator $U\colon X\rightarrow Y$ is absolutely summing.
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