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 Möb of bi-holomorphic automorphisms of the unit disc. A projective unitary representation $$\sigma $$ of Möb is said to be associated with an operator T if $$\varphi (T)= \sigma (\varphi )^* T \sigma (\varphi )$$ for all $$\varphi $$ in Möb. In this paper, we develop a Möbius 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 $$\begin{aligned} {\hat{\sigma }} = (\pi \otimes D_1^+) \oplus \sigma \oplus (\pi _*\otimes D_1^-), \end{aligned}$$ where $$D_1^\pm $$ are two unitary representations (one holomorphic and the other anti-holomorphic) living on the Hardy space $$H^2({\mathbb {D}})$$ , and $$\pi , \pi _*$$ are representations of Möb 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 $$\theta _T(z) = \pi _*(\varphi _z)^* \theta _T(0) \pi (\varphi _z),$$ $$z\in {\mathbb {D}}$$ , where $$\varphi _z$$ is the involution in Möb mapping z to 0. We obtain a concrete realization of this product formula for a large subclass of homogeneous cnu contractions from the Cowen–Douglas class.