3,671 research outputs found
Schur and operator multipliers
Schur multipliers were introduced by Schur in the early 20th century and have
since then found a considerable number of applications in Analysis and enjoyed
an intensive development. Apart from the beauty of the subject in itself,
sources of interest in them were connections with Perturbation Theory, Harmonic
Analysis, the Theory of Operator Integrals and others. Advances in the
quantisation of Schur multipliers were recently made by Kissin and Shulman. The
aim of the present article is to summarise a part of the ideas and results in
the theory of Schur and operator multipliers. We start with the classical Schur
multipliers defined by Schur and their characterisation by Grothendieck, and
make our way through measurable multipliers studied by Peller and Spronk,
operator multipliers defined by Kissin and Shulman and, finally,
multidimensional Schur and operator multipliers developed by Juschenko and the
authors. We point out connections of the area with Harmonic Analysis and the
Theory of Operator Integrals
Completely bounded bimodule maps and spectral synthesis
We initiate the study of the completely bounded multipliers of the Haagerup
tensor product of two copies of the Fourier algebra
of a locally compact group . If is a closed subset of we let
and show that if is a set of
spectral synthesis for then is a set of local
spectral synthesis for . Conversely, we prove that if is a set of
spectral synthesis for and is a Moore group then is a
set of spectral synthesis for . Using the natural
identification of the space of all completely bounded weak* continuous
-bimodule maps with the dual of , we show
that, in the case is weakly amenable, such a map leaves the multiplication
algebra of invariant if and only if its support is contained in
the antidiagonal of .Comment: 44 page
Reduced spectral synthesis and compact operator synthesis
We introduce and study the notion of reduced spectral synthesis, which
unifies the concepts of spectral synthesis and uniqueness in locally compact
groups. We exhibit a number of examples and prove that every non-discrete
locally compact group with an open abelian subgroup has a subset that fails
reduced spectral synthesis. We introduce compact operator synthesis as an
operator algebraic counterpart of this notion and link it with other
exceptional sets in operator algebra theory, studied previously. We show that a
closed subset of a second countable locally compact group satisfies
reduced local spectral synthesis if and only if the subset of satisfies compact operator synthesis. We apply
our results to questions about the equivalence of linear operator equations
with normal commuting coefficients on Schatten -classes.Comment: 43 page
Closable Multipliers
Let (X,m) and (Y,n) be standard measure spaces. A function f in
is called a (measurable) Schur multiplier if
the map , defined on the space of Hilbert-Schmidt operators from
to by multiplying their integral kernels by f, is bounded
in the operator norm.
The paper studies measurable functions f for which 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
- β¦