Scalar perturbations of the Sz.-Nagy—Foias characteristic function

Abstract

When T is a completely non-unitary (cnu) contraction with Sz.-Nagy —Foias characteristic function θ and μ is a scalar (0 < μ < 1) then general theory implies the existence of a cnu contraction T [μ] with characteristic function How to describe T [μ] in terms of T? In this paper, we find a simple answer in case T is in the class C_.0. An operator is called homogeneous if its spectrum is contained in the closed unit disc and all the bi-holomorphic automorphisms of the unit disc lift to automorphisms of the operator T modulo unitary equivalence. When T is homogeneous, so is T [μ] We find explicit formulae for the characteristic functions of the (homogeneous) twisted Bergman shifts – these have product formulae involving the discrete series projective representations of the Möbius group. These formulae lead to an explicit description of the Sz.-Nagy—Foias models of these weighted shifts. Combining our main results with an analytic continuation argument, we find a three-parameter family of homogeneous operators.