Closable Multipliers

Abstract

Let (X, μ) and (Y, ν) be standard measure spaces. A function ${\varphi\in L^\infty(X\times Y,\mu\times\nu)}$ is called a (measurable) Schur multiplier if the map S _φ , defined on the space of Hilbert-Schmidt operators from L ₂(X, μ) to L ₂(Y, ν) by multiplying their integral kernels by φ, is bounded in the operator norm. The paper studies measurable functions φ for which S _φ 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 φ is of Toeplitz type, that is, if φ(x, y) = f(x − y), ${x,y\in G}$, where G is a locally compact abelian group, then the closability of φ is related to the local inclusion of f in the Fourier algebra A(G) of G. If φ is a divided difference, that is, a function of the form (f(x) − f(y))/(x − y), then its closability is related to the “operator smoothness” of the function f. A number of examples of non-closable, norm closable and w*-closable multipliers are presented.