43 research outputs found
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
Backward shift invariant subspaces in the bidisc II
For every invariant subspace NI in the Hardy spaces H2 (f2 ), let Vz and Vw be mulitplication operators on AL Then it is known that the condition Vz V; v;vz on NI holds if and only if J;I is a Demling type invariant subspace. For a backward shift invariant subspace N in H2(f2), two operators Sz and Sw on N are defined by Sz = PN LzPN and Sw = PN Lw PN, where PN is the orthogonal projection from L2(f2) onto N. It is given a characterization of N satisfying szs1:J = s1:JsZ on N
Partial regularity and t-analytic sets for Banach function algebras
In this note we introduce the notion of t-analytic sets. Using this concept, we construct a class of closed prime ideals in Banach function algebras and discuss some problems related to Alling’s conjecture in H infinity. A description of all closed t-analytic sets for the disk-algebra is given. Moreover, we show that some of the assertions in [8] concerning the O-analyticity and S-regularity of certain Banach function algebras are not correct. We also determine the largest set on which a Douglas algebra is pointwise regular
Backward shift invariant subspaces in the bidisc
Suppose that Tq, is a Toeplitz operator with a symbol 1> on the Hardy space H2 on the bidisc. Let N be a backward shift invariant subspace of H2, that is, N is an invariant subspace under r; and T:U. Let P be the orthogonal projection from H2 onto N. For 1> in H00 put Sq, = PTq,[N. In this paper, we give a characterization of a , backward shift invariant subspace which satisfies SzS S Sz
Backward shift invariant subspaces in the bidisc II
For every invariant subspace NI in the Hardy spaces H2 (f2 ), let Vz and Vw be mulitplication operators on AL Then it is known that the condition Vz V; v;vz on NI holds if and only if J;I is a Demling type invariant subspace. For a backward shift invariant subspace N in H2(f2), two operators Sz and Sw on N are defined by Sz = PN LzPN and Sw = PN Lw PN, where PN is the orthogonal projection from L2(f2) onto N. It is given a characterization of N satisfying szs1:J = s1:JsZ on N