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 $C_{\cdot 0}$ with natural positivity assumptions. In particular, we prove that for an $m$-hypercontraction $T \in C_{\cdot 0}$ on a Hilbert space $\mathcal{H}$, there exists a Hilbert space $\mathcal{E}$ and a partially isometric multiplier $\theta \in \mathcal{M}(H^2(\mathcal{E}), A^2_m(\mathcal{H}))$ 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 $A^2_m$ is the weighted Bergman space and $H^2$ is the Hardy space over the unit disc $\mathbb{D}$. We then proceed to study and develop analytic models for doubly commuting $n$-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 $\mathbb{D}^n$.Comment: Revised. 16 pages. To appear in Studia Mathematic