Spaces of operator-valued functions measurable with respect to the strong operator topology

TL;DR

This paper introduces a new space of operator-valued functions measurable in the strong operator topology and characterizes bounded multipliers between certain vector-valued Lp spaces using these functions.

Contribution

It defines the space $L^p[;L(X,Y)]$ of strongly measurable operator-valued functions and provides an isometric characterization of bounded multipliers between vector-valued Lp spaces.

Findings

Functions in $L^p[;L(X,Y)]$ define operator-valued measures with bounded p-variation.

Characterization of bounded multipliers from $L^p(;X)$ to $L^q(;Y)$ using these operator-valued functions.

Establishment of an isometric isomorphism between the space of multipliers and the introduced function space.

Abstract

Let $X$ and $Y$ be Banach spaces and $(\Omega,\Sigma,\mu)$ a finite measure space. In this note we introduce the space $L^p[\mu;L(X,Y)]$ consisting of all (equivalence classes of) functions $\Phi:\Omega \mapsto L(X,Y)$ such that $\omega \mapsto \Phi(\omega)x$ is strongly $\mu$-measurable for all $x\in X$ and $\omega \mapsto \Phi(\omega)f(\omega)$ belongs to $L^1(\mu;Y)$ for all $f\in L^{p'}(\mu;X)$, $1/p+1/p'=1$. We show that functions in $L^p[\mu;\L(X,Y)]$ define operator-valued measures with bounded $p$-variation and use these spaces to obtain an isometric characterization of the space of all $L(X,Y)$-valued multipliers acting boundedly from $L^p(\mu;X)$ into $L^q(\mu;Y)$, $1\le q< p<\infty$.