Isometric factorization of vector measures and applications to spaces of integrable functions

TL;DR

This paper develops an isometric factorization theory for vector measures and their integration operators, refining existing results by establishing isometric equivalences and deepening understanding of the structure of spaces of integrable functions.

Contribution

It introduces an isometric version of the Davis-Figiel-Johnson-Pelczýnski factorization for vector measures, providing sharper results on the structure of $L_1$ spaces associated with these measures.

Findings

Established an isometric factorization framework for vector measures.

Sharpened a result on the structure of $L_1(m)$ spaces with weakly compact integration operators.

Proved that certain $L_1$ spaces are isometrically equivalent to spaces with reflexive Banach space measures.

Abstract

Let $X$ be a Banach space, $\Sigma$ be a $\sigma$-algebra, and $m:\Sigma\to X$ be a (countably additive) vector measure. It is a well known consequence of the Davis-Figiel-Johnson-Pelcz\'{y}nski factorization procedure that there exist a reflexive Banach space $Y$, a vector measure $\tilde{m}:\Sigma \to Y$ and an injective operator $J:Y \to X$ such that $m$ factors as $m=J\circ \tilde{m}$. We elaborate some theory of factoring vector measures and their integration operators with the help of the isometric version of the Davis-Figiel-Johnson-Pelcz\'{y}nski factorization procedure. Along this way, we sharpen a result of Okada and Ricker that if the integration operator on $L_1(m)$ is weakly compact, then $L_1(m)$ is equal, up to equivalence of norms, to some $L_1(\tilde m)$ where $Y$ is reflexive; here we prove that the above equality can be taken to be isometric.