Closable Multipliers

TL;DR

This paper investigates the properties of measurable Schur multipliers, focusing on their closability in different topologies, and explores special cases like Toeplitz functions and divided differences, linking them to harmonic analysis and operator theory.

Contribution

It provides a characterization of w*-closable multipliers, relates norm closability to operator synthesis, and studies special multiplier types with concrete examples.

Findings

Characterization of w*-closable multipliers

Relation of Toeplitz multipliers to Fourier algebra

Connection between divided differences and operator smoothness

Abstract

Let (X,m) and (Y,n) be standard measure spaces. A function f in $L^\infty(X\times Y,m\times n)$ is called a (measurable) Schur multiplier if the map $S_f$, defined on the space of Hilbert-Schmidt operators from $L_2(X,m)$ to $L_2(Y,n)$ by multiplying their integral kernels by f, is bounded in the operator norm. The paper studies measurable functions f for which $S_f$ is closable in the norm topology or in the weak* topology. We obtain a characterisation of w*-closable multipliers and relate the question about norm closability to the theory of operator synthesis. We also study multipliers of two special types: if f is of Toeplitz type, that is, if f(x,y)=h(x-y), x,y in G, where G is a locally compact abelian group, then the closability of f is related to the local inclusion of h in the Fourier algebra A(G) of G. If f is a divided difference, that is, a function of the form (h(x)-h(y))/(x-y), then its closability is related to the "operator smoothness" of the function h. A number of examples of non-closable, norm closable and w*-closable multipliers are presented.