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 $\gamma, |\gamma|=1$. Namely all such unitary perturbations are $U_\gamma:=U+(\gamma-1) (., b_1)_{\mathcal H} b$, where $b\in\mathcal H, \|b\|=1, b_1=U^{-1} b, |\gamma|=1$. For $|\gamma|<1$ operators $U_\gamma$ 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 $U_\gamma$, $|\gamma|<1$ is a completely non-unitary contraction, and thus unitarily equivalent to its functional model $\mathcal M_\gamma$, which is the compression of the multiplication by the independent variable $z$ onto the model space $\mathcal K_{\theta_\gamma}$, where $\theta_\gamma$ is the characteristic function of the contraction $U_\gamma$. The Clark operator $\Phi_\gamma$ is a unitary operator intertwining $U_\gamma, |\gamma|<1$ and its model $\mathcal M_\gamma$, $\mathcal M_\gamma \Phi_\gamma = \Phi_\gamma U_\gamma$. If spectral measure of $U$ is purely singular (equivalently, $\theta_\gamma$ is inner), operator $\Phi_\gamma$ was described from a slightly different point of view by D. Clark. When $\theta_\gamma$ 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 $\Phi^*_\gamma$ 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 $\Phi_\gamma$.