Operator ideal properties of vector measures with finite variation
Given a vector measure $m$ with values in a Banach space $X$, a desirable property (when available) of the associated Banach function space $L^1 (m)$ of all $m$-integrable functions is that $L^1 (m)= L^1(|m|)$, where $|m|$ is the $[0,\infty ]$-valued variation measure of $m$. Closely connected to $m$ is its $X$-valued integration map $ I_m : f \mapsto \int f \, d m $ for $f \in L^1 (m)$. Many traditional operators from analysis arise as integration maps in this way. A detailed study is made of the connection between the property $L^1 (m)= L^1(|m|)$ and the membership of $I_m$ in various classical operator ideals (e.g., the compact, $p$-summing, completely continuous operators). Depending on which operator ideal is under consideration, the geometric nature of the Banach space $X$ may also play a crucial role. Of particular importance in this regard is whether or not $X$ contains an isomorphic copy of the classical sequence space $\ell ^1$. The compact range property of $X$ is also relevant.