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Let      X X       and      Y Y       be real Banach spaces. We prove that an operator      T T       from      X X       into      Y Y       maps compact subsets of      X X       into subsets of      Y Y       that lie inside ranges of      Y Y      -valued measures if and only if its dual operator      T ∗  {T^ * }       factors through a subspace of an      L 1 ( μ  ) {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 ∗  {T^ * }       is      1 1      -summing if and only if      T T       maps compact subsets into subsets of      Y Y       lying inside ranges of vector measures with bounded variation.

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