35 research outputs found
Maximal quadratic modules on *-rings
We generalize the notion of and results on maximal proper quadratic modules
from commutative unital rings to -rings and discuss the relation of this
generalization to recent developments in noncommutative real algebraic
geometry. The simplest example of a maximal proper quadratic module is the cone
of all positive semidefinite complex matrices of a fixed dimension. We show
that the support of a maximal proper quadratic module is the symmetric part of
a prime -ideal, that every maximal proper quadratic module in a
Noetherian -ring comes from a maximal proper quadratic module in a simple
artinian ring with involution and that maximal proper quadratic modules satisfy
an intersection theorem. As an application we obtain the following extension of
Schm\" udgen's Strict Positivstellensatz for the Weyl algebra: Let be an
element of the Weyl algebra which is not negative semidefinite
in the Schr\" odinger representation. It is shown that under some conditions
there exists an integer and elements such
that is a finite sum of hermitian squares. This
result is not a proper generalization however because we don't have the bound
.Comment: 11 page
Operator theory and function theory in Drury-Arveson space and its quotients
The Drury-Arveson space , also known as symmetric Fock space or the
-shift space, is a Hilbert function space that has a natural -tuple of
operators acting on it, which gives it the structure of a Hilbert module. This
survey aims to introduce the Drury-Arveson space, to give a panoramic view of
the main operator theoretic and function theoretic aspects of this space, and
to describe the universal role that it plays in multivariable operator theory
and in Pick interpolation theory.Comment: Final version (to appear in Handbook of Operator Theory); 42 page
Applications of Hilbert Module Approach to Multivariable Operator Theory
A commuting -tuple of bounded linear operators on a
Hilbert space \clh associate a Hilbert module over
in the following sense: where and
. A companion survey provides an introduction to the theory
of Hilbert modules and some (Hilbert) module point of view to multivariable
operator theory. The purpose of this survey is to emphasize algebraic and
geometric aspects of Hilbert module approach to operator theory and to survey
several applications of the theory of Hilbert modules in multivariable operator
theory. The topics which are studied include: generalized canonical models and
Cowen-Douglas class, dilations and factorization of reproducing kernel Hilbert
spaces, a class of simple submodules and quotient modules of the Hardy modules
over polydisc, commutant lifting theorem, similarity and free Hilbert modules,
left invertible multipliers, inner resolutions, essentially normal Hilbert
modules, localizations of free resolutions and rigidity phenomenon.
This article is a companion paper to "An Introduction to Hilbert Module
Approach to Multivariable Operator Theory".Comment: 46 pages. This is a companion paper to arXiv:1308.6103. To appear in
Handbook of Operator Theory, Springe
β-admissibility of observation operators for hypercontractive semigroups
We prove a Weiss conjecture on β -admissibility of observation operators for discrete and continuous γ -hypercontractive semigroups of operators, by representing them in terms of shifts on weighted Bergman spaces and using a reproducing kernel thesis for Hankel operators. Particular attention is paid to the case γ=2 , which corresponds to the unweighted Bergman shift
An Interpolation Problem for Completely Positive Maps on Matrix Algebras: Existence and Solvability
Cataloged from PDF version of article.We present certain existence criteria and parameterizations for an interpolation problem for completely positive maps that take given matrices from a finite set into prescribed matrices. Our approach uses density matrices associated to linear functionals on (Formula presented.) -subspaces of matrices, inspired by the Smith-Ward linear functional and Arveson’s Hahn-Banach Type Theorem. A necessary and sufficient condition for the existence of solutions and a parametrization of the set of all solutions of the interpolation problem in terms of a closed and convex set of an affine space are obtained. Other linear affine restrictions, like trace preserving, can be included as well, hence covering applications to quantum channels that yield certain quantum states at prescribed quantum states. We also perform a careful investigation on the intricate relation between the positivity of the density matrix and the positivity of the corresponding linear functional. © 2014, © 2014 Taylor & Francis