2019/07/09 by Bhaskar Bagchi, Bagchi, Bhaskar, Somnath Hazra +3
Mathematics · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1907.04038
openalex publication_date 2019/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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)^⋆ T σ(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 the two Discrete series representations (one holomorphic and the other anti-holomorphic) living on the Hardy space H2(\mathbb 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∈ \mathbb D, where φz is the involution in Möb mapping z to 0. We obtain a concrete realization of this product formula %the two representations π_⋆ and π for a large subclass of homogeneous cnu contractions from the Cowen-Douglas class.