Closable Multipliers
Artikel i vetenskaplig tidskrift, 2011

Let (X, mu) and (Y, nu) be standard measure spaces. A function. phi is an element of L-infinity(XxY, mu x nu) is called a (measurable) Schur multiplier if the map S phi, defined on the space of Hilbert-Schmidt operators from L-2(X, mu) to L2(Y, nu) by multiplying their integral kernels by phi, is bounded in the operator norm. The paper studies measurable functions phi for which S phi 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 phi is of Toeplitz type, that is, if phi(x, y) = f(x - y), x, y is an element of G, where G is a locally compact abelian group, then the closability of phi is related to the local inclusion of f in the Fourier algebra A(G) of G. If phi 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.

Författare

Victor Shulman

Vologda State University

I. G. Todorov

Queen's University Belfast

Lyudmyla Turowska

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Matematik

Integral Equations and Operator Theory

0378-620X (ISSN) 1420-8989 (eISSN)

Vol. 69 29-62

Fundament

Grundläggande vetenskaper

Ämneskategorier

Matematisk analys

DOI

10.1007/s00020-010-1819-2