74 research outputs found
Operator Theory on Symmetrized Bidisc
A commuting pair of operators (S, P) on a Hilbert space H is said to be a
Gamma-contraction if the symmetrized bidisc is a spectral set of the tuple (S,
P). In this paper we develop some operator theory inspired by Agler and Young's
results on a model theory for Gamma-contractions.
We prove a Beurling-Lax-Halmos type theorem for Gamma-isometries. Along the
way we solve a problem in the classical one-variable operator theory. We use a
"pull back" technique to prove that a completely non-unitary Gamma-contraction
(S, P) can be dilated to a direct sum of a Gamma-isometry and a Gamma-unitary
on the Sz.-Nagy and Foias functional model of P, and that (S, P) can be
realized as a compression of the above pair in the functional model of P.
Moreover, we show that the representation is unique. We prove that a commuting
tuple (S, P) with |S| \leq 2 and |P \leq 1 is a Gamma-contraction if and only
if there exists a compressed scalar operator X with the decompressed numerical
radius not greater than one such that S = X + P X^*. In the commutant lifting
set up, we obtain a unique and explicit solution to the lifting of S where (S,
P) is a completely non-unitary Gamma-contraction. Our results concerning the
Beurling-Lax-Halmos theorem of Gamma-isometries and the functional model of
Gamma-contractions answers a pair of questions of J. Agler and N. J. Young.Comment: 26 pages, revised and final version. To appear in Indiana University
Mathematics Journa
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
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