Related papers: Shift invariant subspaces in the Bloch space
We show that if a nonscalar operator on a separable Hilbert space has a nontrivial invariant subspace, then it has also a nontrivial hyperinvariant subspace. Thus the hyperinvariant subspace problem is equivalent to the invariant subspace…
This article describes Hilbert spaces contractively contained in certain reproducing kernel Hilbert spaces of analytic functions on the open unit disc which are nearly invariant under division by an inner function. We extend Hitt's theorem…
The purpose of this expository note is to give a proof of a Schur-type theorem that characterizes the inner functions in terms of their Taylor coefficients. In view of Beurling's theorem, this provides a sequential characterization of the…
It is known, that every function on the unit sphere in $\bbr^n$, which is invariant under rotations about some coordinate axis, is completely determined by a function of one variable. Similar results, when invariance of a function reduces…
We introduce two kinds of quasi-inner functions. Since every rationally invariant subspace for a shift operator $S_K$ on a vector-valued Hardy space $H^{2}(\Omega,K)$ is generated by a quasi-inner function, we also provide relationships of…
In this paper we present two different problems within the framework of shift-invariant theory. First, we develop a triangular form for shift-preserving operators acting on finitely generated shift-invariant spaces. In case of the normal…
This paper explores the Invariant Subspace Problem in operator theory and functional analysis, examining its applications in various branches of mathematics and physics. The problem addresses the existence of invariant subspaces for bounded…
An algebra of bounded linear operators on a Banach space is said to be {\em strongly compact} if its unit ball is precompact in the strong operator topology, and a bounded linear operator on a Banach space is said to be {\em strongly…
We study the invertibility of Banach algebras elements in their extensions, and invertible extensions of Banach and Hilbert space operators with prescribed growth conditions for the norm of inverses. As applications, the solutions of two…
In this work we provide a characterization of distinct type of (linear and non-linear) maps between Banach spaces in terms of the differentiability of certain class of Lipschitz functions. Our results are stated in an abstract bornological…
The space of Bloch functions on bounded symmetric domains is extended by considering Bloch functions $f$ on the unit ball $B_E$ of finite and infinite dimensional complex Banach spaces in two different ways: by extending the classical Bloch…
In this paper we begin the study of some important Banach spaces of slice hyperholomorphic functions, namely the Bloch, Besov and weighted Bergman spaces, and we also consider the Dirichlet space, which is a Hilbert space. The importance of…
In this paper, we study the cyclicity problem with respect to the forward shift operator $S_b$ acting on the de Branges--Rovnyak space $\mathscr{H}(b)$ associated to a function $b$ in the closed unit ball of $H^\infty$ and satisfying…
Assume that $X$ is a complex separable infinite dimensional Banach space and $\mathcal{B}(X)$ denotes the Banach algebra of all bounded linear operators from $X$ to itself. In 1970, P.R. Halmos raised ten open problems in Hilbert spaces.…
We consider weighted composition operators on spaces of analytic functions on the unit disc, which take values in some complex Banach space. We provide necessary and sufficient conditions for the boundedness and (weak) compactness of…
We introduce and study the following modified version of the Invariant Subspace Problem: whether every operator T on a Banach space has an almost invariant half-space, that is, a subspace Y of infinite dimension and infinite codimension…
By analytic perturbations, we refer to shifts that are finite rank perturbations of the form $M_z + F$, where $M_z$ is the unilateral shift and $F$ is a finite rank operator on the Hardy space over the open unit disc. Here shift refers to…
We present some extensions of classical results that involve elements of the dual of Banach spaces, such as Bishop-Phelp's theorem and James' compactness theorem, but restricting to sets of functionals determined by geometrical properties.…
In the authors' first paper, Beurling-Rudin-Korenbljum type characterization of the closed ideals in a certain algebra of holomorphic functions was used to describe the lattice of invariant subspaces of the shift plus a complex Volterra…
The objective of this article is to study nearly invariant subspaces of the backward shift operator on the real Hardy space. We also investigate nearly invariant subspaces with finite defect, and as a consequence, provide a characterization…