arXiv subjects

Anton D. Baranov

Publications and source records attributed to Anton D. Baranov.

7 recordsLinked to original sources

arXiv2023

Weissler and Bernoulli type inequalities in Bergman spaces

We consider Weissler type inequalities for Bergman spaces with general radial weights and give conditions on the weight $w$ in terms of its moments ensuring that $\|f_r\|_{A^{2n}(w)}\leq \|f\|_{A^2(w)}$ whenever $n\in \mathbb{N}$ and $0< r\le 1/\sqrt{n}$. For noninteger exponents a special case of this inequality is proved which can be considered as a certain analog of the Bernoulli inequality. An example of a monotonic weight is constructed for which these inequalities are no longer true.

math.CV↗

arXiv2022

On Bloch seminorm of finite Blaschke products in the unit disk

We prove that, for any finite Blaschke product $w=B(z)$ in the unit disk, the corresponding Riemann surface over the $w$--plane contains a one-sheeted disk of the radius $0.5$. Moreover, it contains a unit one-sheeted disk with a radial slit. We apply this result to obtain a universal sharp lower estimate of the Bloch seminorm for finite Blaschke products.

math.CV↗

arXiv2016

Completeness and spectral synthesis of nonselfadjoint one-dimensional perturbations of selfadjoint operators

We study spectral properties of nonselfadjoint rank one perturbations of compact selfadjoint operators. The problems under consideration include completeness of eigenvectors, relations between completeness of the perturbed operator and its adjoint, and the spectral synthesis problem. We obtain new criteria for completeness and spectral synthesis in this class as well as a series of counterexamples which show that the spectral structure of rank one perturbations is, in general, unexpectedly rich and complicated. A parallel spectral theory is developed for one-dimensional singular perturbations of unbounded selfadjoint operators. Our approach is based on a functional model for this class which translates the properties of operators to completeness problems for systems of reproducing kernels and their biorthogonals in some spaces of analytic (entire) functions.

math.FA↗

arXiv2016

Completeness of rank one perturbations of normal operators with lacunary spectrum

Suppose $\mathcal{A}$ is a compact normal operator on a Hilbert space $H$ with certain lacunarity condition on the spectrum (which means, in particular, that its eigenvalues go to zero exponentially fast), and let $\mathcal{L}$ be its rank one perturbation. We show that either infinitely many moment equalities hold or the linear span of root vectors of $\mathcal{L}$, corresponding to non-zero eigenvalues, is of finite codimension in $H$. In contrast to classical results, we do not assume that the perturbation is weak.

math.SP↗

arXiv2015

On $L^1$-estimates of derivatives of univalent rational functions

We study the growth of the quantity $\int_{\mathbb{T}}|R'(z)|\,dm(z)$ for rational functions $R$ of degree $n$, which are bounded and univalent in the unit disk, and prove that this quantity may grow as $n^γ$, $γ>0$, when $n\to\infty$. Some applications of this result to problems of regularity of boundaries of Nevanlinna domains are considered. We also discuss a related result by Dolzhenko which applies to general (non-univalent) rational functions.

math.CV↗

arXiv2013

One-dimensional perturbations of unbounded selfadjoint operators with empty spectrum

We study spectral properties of one-dimensional singular perturbations of an unbounded selfadjoint operator and give criteria for the possibility to remove the whole spectrum by a perturbation of this type. A counterpart of our results for the case of bounded operators provides a complete description of compact selfadjoint operators whose rank one perturbation is a Volterra operator.

math.SP↗

arXiv2007

Compact embeddings of model subspaces of the Hardy space

We study embeddings of model (star-invariant) subspaces $K^p_Θ$ of the Hardy space $H^p$, associated with an inner function $Θ$. We obtain a criterion for the compactness of the embedding of $K^p_Θ$ into $L^p(μ)$ analogous to the Volberg--Treil theorem on bounded embeddings and answer a question posed by Cima and Matheson. The proof is based on Bernstein inequalities for functions in $K^p_Θ$. Also we study measures $μ$ such that the embedding operator belongs to a Schatten--von Neumann ideal.

math.CV↗