Related papers: Clark model in general situation

All unitary perturbations of a given unitary operator $U$ by finite rank $d$ operators with fixed range can be parametrized by $(d\times d)$ unitary matrices $\Gamma$; this generalizes unitary rank one ($d=1$) perturbations, where the…

We start with considering rank one self-adjoint perturbations $A_\alpha = A+\alpha(\,\cdot\,,\varphi)\varphi$ with cyclic vector $\varphi\in \mathcal{H}$ on a separable Hilbert space $\mathcal H$. The spectral representation of the…

Given a purely contractive matrix-valued analytic function $\Theta$ on the unit disc $\bm{D}$, we study the $\mc{U} (n)$-parameter family of unitary perturbations of the operator $Z_\Theta$ of multiplication by $z$ in the Hilbert space…

We develop a Clark theory for commuting compressed shift operators on model spaces $K_{\phi}$ associated with inner functions $\phi$ on the bidisk, which exhibits both similarities and marked differences compared to the classical…

We consider rank one perturbations $A_\alpha=A+\alpha(\cdot,\varphi)\varphi$ of a self-adjoint operator $A$ with cyclic vector $\varphi\in\mathcal H_{-1}(A)$ on a Hilbert space $\mathcal H$. The spectral representation of the perturbed…

For a fixed natural number n, we consider a family of rank n unitary perturbations of a completely non-unitary contraction (cnu) with deficiency indices (n,n) on a separable Hilbert space. We relate the unitary dilation of such a…

Let $\mathcal{H}$ be a right quaternionic Hilbert space and let $T$ be a quaternionic normal operator with the domain $\mathcal{D}(T) \subset \mathcal{H}$. Then for a fixed unit imaginary quaternion $m$, there exists a Hilbert basis…

Recently, S. Grivaux showed that there exists a rank one perturbation of a unitary operator in a Hilbert space which is hypercyclic. Another construction was suggested later by the first and the third authors. Here, using a functional model…

Let $U$ be a unitary operator defined on some infinite-dimensional complex Hilbert space ${\cal H}$. Under some suitable regularity assumptions, it is known that a local positive commutation relation between $U$ and an auxiliary…

If $U$ is a unitary operator on a separable complex Hilbert space $\mathcal{H}$, an application of the spectral theorem says there is a conjugation $C$ on $\mathcal{H}$ (an antilinear, involutive, isometry on $\mathcal{H}$) for which $ C U…

Let $B_d$ denote the unit ball of $\mathbb{C}^d$, $d\ge 1$. Given a holomorphic function $\varphi: B_d \to B_1$, we study the corresponding family $\sigma_\alpha[\varphi]$, $\alpha\in\partial B_1$, of Clark measures on the unit sphere…

Weighted discrete Hilbert transforms $(a_n)_n \mapsto \big(\sum_n a_n v_n/(\lambda_j-\gamma_n)\big)_j$ from $\ell^2_v$ to $\ell^2_w$ are considered, where $\Gamma=(\gamma_n)$ and $\Lambda=(\lambda_j)$ are disjoint sequences of points in the…

Let $A$ be a unital operator algebra. Let us assume that every {\it bounded\/} unital homomorphism $u\colon \ A\to B(H)$ is similar to a {\it contractive\/} one. Let $\text{\rm Sim}(u) = \inf\{\|S\|\, \|S^{-1}\|\}$ where the infimum runs…

We extend the basic elements of Clark's theory of rank-one perturbations of backward shifts, to row-contractive operators associated to de Branges-Rovnyak type spaces $\mathcal H(b)$ contractively contained in the Drury-Arveson space on the…

Let $T$ be a self-adjoint operator in a Hilbert space $H$ with domain $\mathcal D(T)$. Assume that the spectrum of $T$ is confined in the union of disjoint intervals $\Delta_k =[\alpha_{2k-1},\alpha_{2k}]$, $k\in \mathbb{Z}$, and $$…

Let $B_n$ denote the unit ball of $\mathbb{C}^n$, $n\ge 1$, and let $\mathcal{D}$ denote a finite product of $B_{n_j}$, $j\ge 1$. Given a non-constant holomorphic function $b: \mathcal{D} \to B_1$, we study the corresponding family…

A classical result of Sz.-Nagy asserts that a Hilbert space contraction operator $T$ can be dilated to a unitary $\cU$. A more general multivariable setting for these ideas is the setup where (i) the unit disk is replaced by a domain…

A commuting tuple of operators $(S_1,\dots, S_{n-1},P)$, defined on a Hilbert space $\mathcal H$, for which the closed symmetrized polydisc \[ \Gamma_n =\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i<j\leq n}z_iz_j,\dots,…

Let $U$ be a unitary operator defined on some infinite-dimensional complex Hilbert space ${\cal H}$. Under some suitable regularity assumptions, it is known that a local positive commutation relation between $U$ and an auxiliary…

We obtain various characterizations of the fundamental operators of $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction and $\Gamma_{E(3; 2; 1, 2)}$-contraction. We also demonstrate some important relations between the fundamental operators of a…