A product formula for homogeneous characteristic functions

Abstract

A bounded linear operator $T$ on a Hilbert space is said to be homogeneous if $\phi (T)$ is unitarily equivalent to $T$ for all $\phi$ 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 $\phi (T)=\sigma (\phi {)}^{\star }T\sigma (\phi )$ for all $\phi$ 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 $$\hat{\sigma }=(\pi \otimes D_1^{+})\oplus \sigma \oplus ({\pi }_{\star }\otimes D_1^{-}),$$ 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 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 ${\theta }_T(z)={\pi }_{\star }({\phi }_z{)}^{\ast }{\theta }_T(0)\pi ({\phi }_z),$ $z\in \mathbb{D}$, where ${\phi }_z$ is the involution in Möb 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.

In this version, some minor errors have been corrected. 33 pages