7,760 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
Method and apparatus for producing an image from a transparent object
The contrast produced from a photographic transparency is controlled by placing the transparency between a pair of partially reflecting mirrors forming walls of an optical cavity. Mirrors trap a collimated laser beam illuminating the transparency so that at least a portion of the beam energy is passed through the transparency plural times. The distance that the light beam travels between the mirrors is controlled as a function of the wavelength of the beam energy to control the phase of light interference in the beam passing through the transparency, thereby controlling the intensity of the beam derived from the mirror downstream of the transparency
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
A unified framework for generalized multicategories
Notions of generalized multicategory have been defined in numerous contexts
throughout the literature, and include such diverse examples as symmetric
multicategories, globular operads, Lawvere theories, and topological spaces. In
each case, generalized multicategories are defined as the "lax algebras" or
"Kleisli monoids" relative to a "monad" on a bicategory. However, the meanings
of these words differ from author to author, as do the specific bicategories
considered. We propose a unified framework: by working with monads on double
categories and related structures (rather than bicategories), one can define
generalized multicategories in a way that unifies all previous examples, while
at the same time simplifying and clarifying much of the theory.Comment: 76 pages; final version, to appear in TA
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
On the low dimensionality of behavioral deficits and alterations of brain network connectivity after focal injury
Sets of multiplicity and closable multipliers on group algebras
We undertake a detailed study of the sets of multiplicity in a second
countable locally compact group and their operator versions. We establish a
symbolic calculus for normal completely bounded maps from the space
of bounded linear operators on into the von
Neumann algebra of and use it to show that a closed subset
is a set of multiplicity if and only if the set is a set of operator multiplicity.
Analogous results are established for -sets and -sets. We show that
the property of being a set of multiplicity is preserved under various
operations, including taking direct products, and establish an Inverse Image
Theorem for such sets. We characterise the sets of finite width that are also
sets of operator multiplicity, and show that every compact operator supported
on a set of finite width can be approximated by sums of rank one operators
supported on the same set. We show that, if satisfies a mild approximation
condition, pointwise multiplication by a given measurable function defines a closable multiplier on the reduced C*-algebra
of if and only if Schur multiplication by the function , given by , is a closable
operator when viewed as a densely defined linear map on the space of compact
operators on . Similar results are obtained for multipliers on .Comment: 51 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
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