A product formula for homogeneous characteristic functions

Abstract

A bounded linear operator T on a Hilbert space is said to be homogeneous if φ(T) is unitarily equivalent to T for all φ in the group Möb of bi-holomorphic automorphisms of the unit disc. A projective unitary representation σ of Möb is said to be associated with an operator T if φ(T)=σ(φ)∗Tσ(φ) for all φ 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 σ, then there is a unique projective unitary representation σ^, extending σ, associated with the minimal unitary dilation of T. The representation σ^ is given in terms of σ by the formula σ^=(π⊗D+1)⊕σ⊕(π∗⊗D−1), where D±1 are two unitary representations (one holomorphic and the other anti-holomorphic) living on the Hardy space H2(D), and π,π∗ are representations of Möb living on the two defect spaces of T defined explicitly in terms of σ. Moreover, a cnu contraction T has an associated representation if and only if its Sz.-Nagy-Foias characteristic function θT has the product form θT(z)=π∗(φz)∗θT(0)π(φz), z∈D, where φz 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.