2,490 research outputs found
Topological radicals, V. From algebra to spectral theory
We introduce and study procedures and constructions in the theory of the
joint spectral radius that are related to the spectral theory. In particular we
devlop the theory of the scattered radical. Among applications we find some
sufficient conditions of continuity of the spectrum and spectral radii of
various types, and prove that in GCR C*-algebras the joint spectral radius is
continuous on precompact subsets and coincides with the Berger-Wang radius
Topological radicals, II. Applications to spectral theory of multiplication operators
We develop the spectral radius technique and the theory of tensor radicals.
As applications we obtain numerous results on mutiplication operators in Banach
algebras and Operator bimodules
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
Some remarks on invariant subspaces in real Banach spaces (revised version)
It is proved that a commutative algebra of operators on a reflexive real
Banach space has an invariant subspace if each operator satisfies the
condition where is the essential norm. This
implies the existence of an invariant subspace for every commutative family of
essentially selfadjoint operators on a real Hilbert space
On the spectrum of multiplication operators
We study relations between spectra of two operators that are connected to
each other through some intertwining conditions. As application we obtain new
results on the spectra of multiplication operators on B(\cl H) relating it to
the spectra of the restriction of the operators to the ideal of
Hilbert-Schmidt operators. We also solve one of the problems, posed in
[B.Magajna, Proc. Amer. Math. Soc, 141 2013, 1349-1360] about the positivity of
the spectrum of multiplication operators with positive operator coefficients
when the coefficients on one side commute. Using the Wiener-Pitt phenomena we
show that the spectrum of a multiplication operator with normal coefficients
satisfying the Haagerup condition might be strictly larger than the spectrum of
its restriction to .Comment: 12 pages, v2: corrected some typos, to appear in Methods of Funct.
Anal. Topo
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