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.Comment: 35 page

Shulman, V. S.

Todorov, I. G.

Turowska, L.

English

arXiv

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

Shulman, Victor

Turowska, Lyudmila

Chalmers Publication Library

Closable Multipliers

Turowska, Lyudmyla

Chalmers Research

V. S. Shulman

I. G. Todorov

L. Turowska

CiteSeerX

CLOSABLE MULTIPLIERS

Let $(X,\mu)$ and $(Y,\nu)$ be standard measure spaces. A function $\nph\in L^\infty(X\times Y,\mu\times\nu)$ is called a (measurable) Schur multiplier if the map $S_\nph$, defined on the space of Hilbert-Schmidt operators from $L_2(X,\mu)$ to $L_2(Y,\nu)$ by multiplying their integral kernels by $\nph$, is bound-ed in the operator norm. The paper studies measurable functions $\nph$ for which $S_\nph$ 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 $\nph$ is of Toeplitz type, that is, if $\nph(x,y)=f(x-y)$, $x,y\in G$, where $G$ is a locally compact abelian group, then the closability of $\nph$ is related to the local inclusion of $f$ in the Fourier algebra $A(G)$ of $G$. If $\nph$ 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

Shulman, V.S.

Todorov, Ivan

Queen's University Belfast Research Portal

Closable multipliers

Closable Multipliers

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