11 research outputs found
Operator Positivity and Analytic Models of Commuting Tuples of Operators
We study analytic models of operators of class with natural
positivity assumptions. In particular, we prove that for an
-hypercontraction on a Hilbert space ,
there exists a Hilbert space and a partially isometric multiplier
such that
\mathcal{H} \cong \mathcal{Q}_{\theta} = A^2_m(\mathcal{H}) \ominus \theta
H^2(\mathcal{E}), \quad \quad \mbox{and} \quad \quad T \cong
P_{\mathcal{Q}_{\theta}} M_z|_{\mathcal{Q}_{\theta}},where is the
weighted Bergman space and is the Hardy space over the unit disc
. We then proceed to study and develop analytic models for doubly
commuting -tuples of operators and investigate their applications to joint
shift co-invariant subspaces of reproducing kernel Hilbert spaces over
polydisc. In particular, we completely analyze doubly commuting quotient
modules of a large class of reproducing kernel Hilbert modules, in the sense of
Arazy and Englis, over the unit polydisc .Comment: Revised. 16 pages. To appear in Studia Mathematic
Commuting row contractions with polynomial characteristic functions
A characteristic function is a special operator-valued analytic function defined on the open unit ball of C n associated with an n-tuple of commuting row contraction on some Hilbert space. In this paper, we continue our study of the representations of n-tuples of commuting row contractions on Hilbert spaces, which have polynomial characteristic functions. Gleason’s problem plays an important role in the representations of row contractions. We further complement the representations of our row contractions by proving theorems concerning factorizations of characteristic functions. We also emphasize the importance and the role of noncommutative operator theory and noncommutative varieties to the classification problem of polynomial characteristic functions
\clw-hypercontractions and their model
We revisit the study of -hypercontractions corresponding to a single
weight sequence introduced by Olofsson in
\cite{O} and find an analogue of Nagy-Foias characteristic function in this
setting. Explicit construction of characteristic functions is obtained and it
is shown to be a complete unitary invariant. By considering a multi-weight
sequence \clw and \clw-hypercontractions we extend Olofsson's work \cite{O}
in the multi-variable setting. Model for \clw-hypercontractions is obtained
by finding their dilations on certain weighted Bergman spaces over the polydisc
corresponding to the multi-weight sequence \clw. This recovers and provides a
different proof of the earlier work of Curto and Vasilescu \cite{CVPoly, CV}
for -contractive multi-operators through a particular choice of
multi-weight sequence.Comment: 31 pages, updated version, comments are welcom