Related papers: Operators invariant relative to a completely nonun…

We consider operators acting on a Hilbert space that can be written as the sum of a shift and a diagonal operator and determine when the operator is hyponormal. The condition is presented in terms of the norm of an explicit block Jacobi…

The truncated Hilbert transform with overlap $H_T$ is an operator that arises in tomographic reconstruction from limited data, more precisely in the method of Differentiated Back-Projection (DBP). Recent work [1] has shown that the singular…

Let $S$ be a symmetric operator with equal defect numbers and let $\mathfrak{U}$ be a set of unitary operators in a Hilbert space $\mathfrak{H}$. The operator $S$ is called $\mathfrak{U}$-invariant if $US=SU$ for all $U\in\mathfrak{U}$.…

Using the tools of Sz.-Nagy--Foias theory of contractions, we describe in detail the invariant subspaces of the operator $ S\oplus S^* $, where $ S $ is the unilateral shift on a Hilbert space. This answers a question of C\^amara and Ross.

The famous Lomonosov's invariant subspace theorem states that if a continuous linear operator T on an infinite-dimensional normed space E "commutes" with a compact nonzero operator K, i.e., TK=KT, then T has a non-trivial closed invariant…

In this paper we study sufficient conditions for an operator to have an almost-invariant half-space. As a consequence, we show that if $X$ is an infinite-dimensional complex Banach space then every operator $T\in\mathcal{L}(X)$ admits an…

We study the asymptotic behaviour of contractive operators and strongly continuous semigroups on separable Hilbert spaces using the notion of rigidity. In particular, we show that a "typical" contraction $T$ contains the unit circle times…

Let $A$ be a non-negative self-adjoint operator in a Hilbert space $\mathcal{H}$ and $A_{0}$ be some densely defined closed restriction of $A_{0}$, $A_{0}\subseteq A \neq A_{0}$. It is of interest to know whether $A$ is the unique…

In this paper we obtain a complete characterization of reducing, invariant, and hyperinvariant subspaces for the completely non-unitary component of a power partial isometry. In particular, precise characterization of reducing, invariant,…

We consider Toeplitz operators defined on a concave corner-shaped subset of the square lattice. We obtain a necessary and sufficient condition for these operators to be Fredholm. We further construct a Fredholm concave corner Toeplitz…

For a power bounded or polynomially bounded operator $T$ sufficient conditions for the existence of a nontrivial hyperinvariant subspace are given. The obtained hyperinvariant subspaces of $T$ have the form of the closure of the range of…

We prove the spectral mapping theorem $\sigma_e(A_\phi) = \phi(\sigma_e(A_z))$ for the Fredholm spectrum of a truncated Toeplitz operator $A_\phi$ with symbol $\phi$ in the Sarason algebra $C+H^\infty$ acting on a coinvariant subspace…

Given a countable dense subset D of an infinite-dimensional separable Hilbert space H the set of operators for which every vector in D except zero is hypercyclic (weakly supercyclic) is residual for the strong (resp. weak) operator topology…

Let $T$ be a self-adjoint operator on a finite dimensional Hilbert space. It is shown that the distribution of the eigenvalues of a compression of $T$ to a subspace of a given dimension is almost the same for almost all subspaces. This is a…

Given a commuting d-tuple $\bar T=(T_1,...,T_d)$ of otherwise arbitrary nonnormal operators on a Hilbert space, there is an associated Dirac operator $D_{\bar T}$. Significant attributes of the d-tuple are best expressed in terms of…

Let $\gH$ be a Hilbert space and let $A$ be a simple symmetric operator in $\gH$ with equal deficiency indices $d:=n_\pm(A)<\infty$. We show that if, for all $\l$ in an open interval $I\subset\bR$, the dimension of defect subspaces…

We establish the following result, confirming a conjecture of Jean Esterle. For each closed subset $E$ of the unit circle of Lebesgue measure zero, there exists a positive sequence $u_n\to\infty$ with the following property: if $T$ is a…

If T is a bounded linear operator acting on an infinite-dimensional Banach space, then there exists and operator F of rank at most one and arbitrarily small norm such that T-F has an invariant subspace of infinite dimension and codimension.…

An algebra A of operators on a Banach space X is called strictly semi-transitive if for all non-zero x,y in X there exists an operator S in A such that Sx=y or Sy=x. We show that if A is norm-closed and strictly semi-transitive, then every…

For $0<r<1$, let us consider the following annulus: \[ \mathbb A_r= \{ z\in \mathbb C\, : \, r<|z|<1 \}. \] A Hilbert space operator $T$ for which $\overline{\mathbb A}_r$ is a spectral set is called an $\mathbb A_r$-\textit{contraction}.…