Clark model in general situation

Abstract

For a unitary operator the family of its unitary perturbations by rank one operators with fixed range is parametrized by a complex parameter $γ, |γ|=1$. Namely all such unitary perturbations are $U_γ:=U+(γ-1) (., b_1)_{\mathcal H} b$, where $b\in\mathcal H, \|b\|=1, b_1=U^{-1} b, |γ|=1$. For $|γ|<1$ operators $U_γ$ are contractions with one-dimensional defects. Restricting our attention on the non-trivial part of perturbation we assume that $b$ is cyclic for $U$. Then the operator $U_γ$, $|γ|<1$ is a completely non-unitary contraction, and thus unitarily equivalent to its functional model $\mathcal M_γ$, which is the compression of the multiplication by the independent variable $z$ onto the model space $\mathcal K_{θ_γ}$, where $θ_γ$ is the characteristic function of the contraction $U_γ$. The Clark operator $Φ_γ$ is a unitary operator intertwining $U_γ, |γ|<1$ and its model $\mathcal M_γ$, $\mathcal M_γΦ_γ= Φ_γU_γ$. If spectral measure of $U$ is purely singular (equivalently, $θ_γ$ is inner), operator $Φ_γ$ was described from a slightly different point of view by D. Clark. When $θ_γ$ is extreme point of the unit ball in $H^\infty$ was treated by D. Sarason using the sub-Hardy spaces introduced by L. de Branges. We treat the general case and give a systematic presentation of the subject. We find a formula for the adjoint operator $Φ^*_γ$ which is represented by a singular integral operator, generalizing the normalized Cauchy transform studied by A. Poltoratskii. We present a "universal" representation that works for any transcription of the functional model. We then give the formulas adapted for the Sz.-Nagy--Foias and de Branges--Rovnyak transcriptions, and finally obtain the representation of $Φ_γ$.