Operator Theory on Symmetrized Bidisc

TL;DR

This paper develops operator theory for Gamma-contractions on Hilbert spaces, including dilation, model theory, and a Beurling-Lax-Halmos theorem, extending classical results and answering open questions in the field.

Contribution

It introduces a dilation and model theory for Gamma-contractions, including a Beurling-Lax-Halmos theorem for Gamma-isometries, and provides explicit solutions in the commutant lifting setting.

Findings

Proved a Beurling-Lax-Halmos type theorem for Gamma-isometries.

Established dilation results for non-unitary Gamma-contractions.

Provided explicit solutions for lifting problems in the Gamma-contraction setting.

Abstract

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.