Operator Theory on Symmetrized Bidisc

Abstract

A commuting pair of operators (S, P ) on a Hilbert space H is said to be a Γ-contraction if the symmetrized bidisc Γ = {(z1 + z2, z1z2) : |z1|, |z2| ≤ 1} 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 Γ-contractions. We prove a Beurling-Lax-Halmos type theorem for Γ-isometries. Along the way we solve a problem in the classical one-variable operator theory, namely, a non-zero Mz-invariant subspace S of H E∗(D) is invariant under the analytic Toeplitz operator with the operatorvalued polynomial symbol p(z) = A + A∗z if and only if the Beurling-Lax-Halmos inner multiplier Θ of S satisfies (A+A∗z)Θ = Θ(B +B∗z), for some unique operator B. We use a ”pull back” technique to prove that a completely non-unitary Γ-contraction (S, P ) can be dilated to a pair (((A+AMz)⊕ U), (Mz ⊕Meit)), which is the direct sum of a Γ-isometry and a Γ-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 QP of P as (PQP ((A+A Mz)⊕ U)|QP , PQP (Mz ⊕Meit)|QP ). Moreover, we show that this representation is unique. We identify a complete set of unitary invariants for the class of completely non-unitary Γ-contractions. We prove that a commuting tuple (S, P ) with ∥S∥ ≤ 2 and ∥P∥ ≤ 1 is a Γ-contraction if and only if there exists a bounded linear operator X such that S = X +X∗P and both X and X∗ commutes with P and the numerical radius w(X) ≤ 1. 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 Γ-contraction. Our results concerning the Beurling-Lax-Halmos theorem of Γ-isometries and the functional model of Γ-contractions answers a pair of questions of J. Agler and N. J. Young.