Operators on Banach spaces taking compact sets inside ranges of vector measures

Abstract:

Let $X$ and $Y$ be real Banach spaces. We prove that an operator $T$ from $X$ into $Y$ maps compact subsets of $X$ into subsets of $Y$ that lie inside ranges of $Y$-valued measures if and only if its dual operator ${T^ * }$ factors through a subspace of an ${L^1}(\mu )$-space. In fact, we prove that every compact is taken into a subset of a compact range. We also prove that ${T^ * }$ is $1$-summing if and only if $T$ maps compact subsets into subsets of $Y$ lying inside ranges of vector measures with bounded variation.