Clark model in the general situation

Abstract

For a unitary operator U in a Hilbert space H the family of its unitary perturbations by rank one operators with fixed range is parametrized by a complex parameter γ, ǀγǀ = 1. Namely, all such unitary perturbations are operators U _γ := U + (γ − 1)( ·, b ₁) _H b, where b ∈ H, ǁbǁ = 1, b ₁ = U ⁻¹ b, ǀγǀ = 1. For ǀγǀ < 1, the operators U _γ are contractions with one-dimensional defects.

Restricting our attention to the non-trivial part of perturbation, we assume that b is a cyclic vector for U, i.e., ${\rm H} = \overline {span} \left\{ {{U^n}b:n \in \mathbb{Z}} \right\}$. In this case, the operator U _γ, ǀγǀ < 1 is a completely non-unitary contraction and thus unitarily equivalent to its functional model M _γ , which is the compression of the multiplication by the independent variable z onto the model space ${K_{{\theta _\gamma }}}$; here, θ _γ is the characteristic function of the contraction U _γ .

The Clark operator Φ _γ is a unitary operator intertwining the operator U _γ , ǀγǀ < 1 (in the spectral representation of the operator U) and its model M _γ , M _γ Φ _γ = Φ _γ U _γ . In the case when the spectral measure of U is purely singular (equivalently, the characteristic function θ _γ is inner), the operator Φ _γ was described from a slightly different point of view by D. Clark [3]. The case where θ _γ is an extreme point of the unit ball in H ^∞ was treated by D. Sarason [18], using the sub-Hardy spaces H(θ) introduced by L. de Branges.

In this paper, we treat the general case and give a systematic presentation of the subject. We first find a formula for the adjoint operator Φ ^* _γ , which is represented by a singular integral operator, generalizing in a sense the normalized Cauchy transform studied by A. Poltoratskii [16]. We begin by presenting a “universal” representation that works for any transcription of the functional model. We then give the formulas adapted for specific transcriptions of the model, such as Sz.-Nagy–Foiaş and the de Branges–Rovnyak transcriptions. Finally, we obtain the representation of Φ _γ .