A Product Formula for Homogeneous Characteristic Functions

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⊗D1+)⊕σ⊕(π∗⊗D1-), where D1± are two unitary representations (one holomorphic and the other anti-holomorphic) living on the Hardy space H²(D) , and π, π<inf>∗</inf> 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 θ<inf>T</inf> has the product form θT(z)=π∗(φz)∗θT(0)π(φz),z∈ D , where φ<inf>z</inf> 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.