Compactness of the integration operator associated with a vector measure
Tom 150 / 2002
Studia Mathematica 150 (2002), 133-149 MSC: Primary 28B05, 46G10, 47B05. DOI: 10.4064/sm150-2-3
A characterization is given of those Banach-space-valued vector measures $m$ with finite variation whose associated integration operator $I_m:f \mapsto \int f \kern .16667em dm$ is compact as a linear map from $L^1(m)$ into the Banach space. Moreover, in every infinite-dimensional Banach space there exist nontrivial vector measures $m$ (with finite variation) such that $I_m$ is compact, and other $m$ (still with finite variation) such that $I_m$ is not compact. If $m$ has infinite variation, then $I_m$ is never compact.