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Anton Baranov

Publications and source records attributed to Anton Baranov.

At least 37 records · Page 2Linked to original sources
arXiv2018

Spectral theory of rank one perturbations of normal compact operators

We construct a functional model for rank one perturbations of compact normal operators acting in a certain Hilbert spaces of entire functions generalizing de Branges spaces. Using this model we study completeness and spectral synthesis problems for such perturbations. Previously, we developed the spectral theory of rank one perturbations in the selfadjoint case. In the present paper we extend and significantly simplify most of known results in the area. We also prove an Ordering Theorem for invariant subspaces with common spectral part. This result is essentially new even for rank one perturbations of compact selfadjoint operators.

math.FA↗
arXiv2018

Synthesable differentiation-invariant subspaces

We describe differentiation-invariant subspaces of $C^\infty(a,b)$ which admit spectral synthesis. This gives a complete answer to a question posed by A.~Aleman and B.~Korenblum. It turns out that this problem is related to a classical problem of approximation by polynomials on the real line. We will depict an intriguing connection between these problems and the theory of de Branges spaces.

math.CV↗
arXiv2018

Krein-type theorems and ordered structure for Cauchy-de Branges spaces

We extend some results of M.G. Krein to the class of entire functions which can be represented as ratios of discrete Cauchy transforms in the plane. As an application we obtain new versions of de Branges' Ordering Theorem for nearly invariant subspaces in a class of Hilbert spaces of entire functions. Examples illustrating sharpness of the obtained results are given.

math.CV↗
arXiv2018

De Branges spaces and Fock spaces

Relations between two classes of Hilbert spaces of entire functions, de Branges spaces and Fock-type spaces with non-radial weights, are studied. It is shown that any de Branges space can be realized as a Fock-type space with equivalent area norm, and several constructions of a representing weight are suggested. For some special classes of weights (e.g., weights depending on the imaginary part only) the corresponding de Branges spaces are explicitly described.

math.CV↗
arXiv2017

On hypercyclic rank one perturbations of unitary operators

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 for rank one perturbations of singular unitary operators, we give yet another construction of hypercyclic rank one perturbation of a unitary operator. In particular, we show that any Carleson set on the circle can be the spectrum of a perturbed (hypercyclic) operator.

math.FA↗
arXiv2017

Univalent functions in model spaces: revisited

Motivated by a problem in approximation theory, we find a necessary and sufficient condition for a model (backward shift invariant) subspace $K_\varTheta = H^2\ominus \varTheta H^2$ of the Hardy space $H^2$ to contain a bounded univalent function.

math.CV↗
arXiv2017

The Young type theorem in weighted Fock spaces

We prove that for every radial weighted Fock space, the system biorthogonal to a complete and minimal system of reproducing kernels is also complete under very mild regularity assumptions on the weight. This result generalizes a theorem by Young on reproducing kernels in the Paley--Wiener space and a recent result of Belov for the classical Bargmann--Segal--Fock space.

math.CV↗
arXiv2016

Hypercyclic Toeplitz operators

We study hypercyclicity of the Toeplitz operators in the Hardy space $H^2(\mathbb{D})$ with symbols of the form $p(\bar{z}) +ϕ(z)$, where $p$ is a polynomial and $ϕ\in H^\infty(\mathbb{D})$. We find both necessary and sufficient conditions for hypercyclicity which almost coincide in the case when ${\rm deg}\, p =1$.

math.FA↗
arXiv2016

Boundedness of the differentiation operator in model spaces and application to Peller type inequalities

Given an inner function $Θ$ in the unit disc $\mathbb{D}$, we study the boundedness of the differentiation operator which acts from from the model subspace $K\_Θ=\left(ΘH^{2}\right)^{\perp}$ of the Hardy space $H^{2},$ equiped with the $BMOA$-norm, to some radial-weighted Bergman space. As an application, we generalize Peller's inequality for Besov norms of rational functions $f$ of degree $n\geq1$ having no poles in the closed unit disc $\overline{\mathbb{D}}$.

math.FA↗
arXiv2015

De Branges functions of Schroedinger equations

We characterize the Hermite-Biehler (de Branges) functions $E$ which correspond to Shroedinger operators with $L^2$ potential on the finite interval. From this characterization one can easily deduce a recent theorem by Horvath. We also obtain a result about location of resonances.

math.CV↗
arXiv2015

Geometry of reproducing kernels in model spaces near the boundary

We study two geometric properties of reproducing kernels in model spaces $K\_θ$where $θ$ is an inner function in the disc: overcompleteness and existence of uniformly minimalsystems of reproducing kernels which do not contain Riesz basic sequences. Both of these properties are related to the notion of the Ahern--Clark point. It is shown that "uniformly minimal non-Riesz"$ $ sequences of reproducing kernelsexist near each Ahern--Clark point which is not an analyticity point for $θ$, whileovercompleteness may occur only near the Ahern--Clark points of infinite orderand is equivalent to a "zero localization property". In this context the notion ofquasi-analyticity appears naturally, and as a by-product of our results we give conditions in thespirit of Ahern--Clark for the restriction of a model space to a radius to be a class ofquasi-analyticity.

math.CV↗
arXiv2015

Gap Theorem for Separated Sequences without Pain

We give a simple and straightforward proof of the Gap Theorem for separated sequences by A. Poltoratski and M. Mitkovski using the Beurling--Malliavin formula for the radius of completeness.

math.CV↗
arXiv2015

Spectral synthesis in de Branges spaces

We solve completely the spectral synthesis problem for reproducing kernels in the de Branges spaces $\mathcal{H}(E)$. Namely, we describe the de Branges spaces $\mathcal{H}(E)$ such that all $M$-bases of reproducing kernels (i.e., complete and minimal systems $\{k_λ\}_{λ\inΛ}$ with complete biorthogonal $\{g_λ\}_{λ\inΛ}$) are strong $M$-bases (i.e., every mixed system $\{k_λ\}_{λ\inΛ\setminus\tilde Λ} \cup\{g_λ\}_{λ\in \tilde Λ}$ is also complete). Surprisingly this property takes place only for two essentially different classes of de Branges spaces: spaces with finite spectral measure and spaces which are isomorphic to Fock-type spaces of entire functions. The first class goes back to de Branges himself, the second class appeared in a recent work of A. Borichev and Yu. Lyubarskii. Moreover, we are able to give a complete characterisation of this second class in terms of the spectral data for $\mathcal{H}(E)$. In addition, we obtain some results about possible codimension of mixed systems for a fixed de Branges space $\mathcal{H}(E)$, and prove that any minimal system of reproducing kernels in $\mathcal{H}(E)$ is contained in an exact system of reproducing kernels.

math.CV↗
arXiv2014

Sampling, interpolation and Riesz bases in small Fock spaces

We give a complete description of Riesz bases of reproducing kernels in small Fock spaces. This characterization is in the spirit of the well known Kadets--Ingham 1/4 theorem for Paley--Wiener spaces. Contrarily to the situation in Paley--Wiener spaces, a link can be established between Riesz bases in the Hilbert case and corresponding complete interpolating sequences in small Fock spaces with associated uniform norm. These results allow to show that if a sequence has a density stricly different from the critical one then either it can be completed or reduced to a complete interpolating sequence. In particular, this allows to give necessary and sufficient conditions for interpolation or sampling in terms of densities.

math.CV↗
arXiv2014

Localization of zeros for Cauchy transforms

We study the localization of zeros of Cauchy transforms of discrete measures on the real line. This question is motivated by the theory of canonical systems of differential equations. In particular, we prove that the spaces of Cauchy transforms having the localization property are in one-to-one correspondence with the canonical systems of special type, namely, those whose Hamiltonians consist only of indivisible intervals accumulating on the left. Various aspects of the localization phenomena are studied in details. Connections with the density of polynomials and other topics in analysis are discussed.

math.CV↗
arXiv2013

Subspaces of $C^\infty$ invariant under the differentiation

Let $L$ be a proper differentiation invariant subspace of $C^\infty(a,b)$ such that the restriction operator $\frac{d}{dx}\bigl{|}_L$ has a discrete spectrum $Λ$ (counting with multiplicities). We prove that $L$ is spanned by functions vanishing outside some closed interval $I\subset(a,b)$ and monomial exponentials $x^ke^{λx}$ corresponding to $Λ$ if its density does not exceed the critical value $\frac{|I|}{2π}$, and moreover, we show that the result is not necessarily true when the density of $Λ$ equals the critical value. This answers a question posed by the first author and B. Korenblum. Finally, if the residual part of $L$ is trivial, then $L$ is spanned by the monomial exponentials it contains.

math.CV↗