IUMJ

Title: Operator Theory on Symmetrized Bidisc

Authors: Jaydeb Sarkar

Issue: Volume 64 (2015), Issue 3, 847-873

Abstract:

A commuting pair of operators $(S,P)$ on a Hilbert space $\mathcal{H}$ is said to be a $\Gamma$-contraction if the symmetrized bidisc \[ \Gamma=\{(z_1+z_2,z_1z_2):|z_1|,|z_2|\leq1\} \] 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: namely, a non-zero $M_z$-invariant subspace $\mathcal{S}$ of $H^2_{\mathcal{E}_{*}}(\mathbb{D})$ is invariant under the analytic Toeplitz operator with the operator-valued polynomial symbol $p(z)=A+A^{*} z$ if and only if the Beurling-Lax-Halmos inner multiplier $\Theta$ of $\mathcal{S}$ satisfies $(A+A^{*}z)\Theta=\Theta(B+B^{*}z)$, for some unique operator $B$. We use a "pull back" technique to prove that a completely non-unitary $\Gamma$-contraction $(S, P)$ can be dilated to a pair \[ (((A+A^{*}M_z)\oplus U), (M_z\oplus M_{e^{it}})), \] which is the direct sum of a $\Gamma$-isometry and a $\Gamma$-unitary on the Sz.-Nagy and Foias functional model of $P$, and to prove that $(S, P)$ can be realized as a compression of the above pair in the functional model $\mathcal{Q}_P$ of $P$ as \[ \big(\bm{P}_{\mathcal{Q}_P}((A+A^{*}M_z)\oplus U)\big|_{\mathcal{Q}_P} \bm{P}_{\mathcal{Q}_P}(M_z\oplus M_{e^{it}})\big|_{\mathcal{Q}_P}\big). \] Moreover, we show that this representation is unique. We prove that a commuting tuple $(S,P)$ with $\|S\|\leq2$ and $\|P\|\leq1$ 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 setup, 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 answer a pair of questions of J.\ Agler and N.\:J.\ Young.