Operator Theory on Symmetrized Bidisc

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 $H^2_{\mathcal{E}_{*}}(\mathbb{D})$ 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 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 $(((A+A^*M_z)\oplus U), (M_z \oplus M_{e^{it}}))$, which is the direct sum of a Γ-isometry and a Γ-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 QP of P as $(\mathbf{P}{_\mathcal{Q_{P}}}((A+A^*M_z)\oplus U)|{_\mathcal{Q_{P}}},\mathbf{P}{_\mathcal{Q_{P}}}(M_z\oplus M_{e^{it}})|{_\mathcal{Q_{P}}})$. Moreover, we show that this representation is unique. 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 compressed scalar operator X with the decompressed numerical radius not greater than one, such that S = X + PX*. 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 Γ-contraction. Our results concerning the Beurling-Lax-Halmos theorem of Γ-isometries and the functional model of Γ-contractions answer a pair of questions of J. Agler and N. J. Young.