A product formula for homogeneous characteristic functions

Abstract

A bounded linear operator $T$ on a Hilbert space is said to be homogeneous if $\varphi(T)$ is unitarily equivalent to $T$ for all $\varphi$ in the group M\"{o}b of bi-holomorphic automorphisms of the unit disc. A projective unitary representation $\sigma$ of M\"{o}b is said to be associated with an operator T if $\varphi(T)= \sigma(\varphi)^\star T \sigma(\varphi)$ for all $\varphi$ in M\"{o}b. In this paper, we develop a M\"{o}bius equivariant version of the Sz.-Nagy--Foias model theory for completely non-unitary (cnu) contractions. As an application, we prove that if T is a cnu contraction with associated (projective unitary) representation $\sigma$, then there is a unique projective unitary representation $\hat{\sigma}$, extending $\sigma$, associated with the minimal unitary dilation of $T$. The representation $\hat{\sigma}$ is given in terms of $\sigma$ by the formula $$ \hat{\sigma} = (\pi \otimes D_1^+) \oplus \sigma \oplus (\pi_\star \otimes D_1^-), $$ where $D_1^\pm$ are the two Discrete series representations (one holomorphic and the other anti-holomorphic) living on the Hardy space $H^2(\mathbb D)$, and $\pi, \pi_\star$ are representations of M\"{o}b living on the two defect spaces of $T$ defined explicitly in terms of $\sigma$. Moreover, a cnu contraction $T$ has an associated representation if and only if its Sz.-Nagy--Foias characteristic function $\theta_T$ has the product form $\theta_T(z) = \pi_\star(\varphi_z)^* \theta_T(0) \pi(\varphi_z),$ $z\in \mathbb D$, where $\varphi_z$ is the involution in M\"{o}b mapping $z$ to $0.$ We obtain a concrete realization of this product formula %the two representations $\pi_\star$ and $\pi$ for a large subclass of homogeneous cnu contractions from the Cowen-Douglas class.