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Stanislas Kupin

Publications and source records attributed to Stanislas Kupin.

6 recordsLinked to original sources
arXiv2024

On the growth of resolvent of Toeplitz operators

We study the growth of the resolvent of a Toeplitz operator $T_b$, defined on the Hardy space, in terms of the distance to its spectrum $σ(T_b)$. We are primarily interested in the case when the symbol $b$ is a Laurent polynomial (\emph{i.e., } the matrix $T_b$ is banded). We show that for an arbitrary such symbol the growth of the resolvent is quadratic, and under certain additional assumption it is linear. We also prove the quadratic growth of the resolvent for a certain class of non-rational symbols.

math.SP↗
arXiv2022

On discrete spectra of Bergman--Toeplitz operators with harmonic symbols

In the present article, we study the discrete spectrum of certain bounded Toeplitz operators with harmonic symbol on a Bergman space. Using the methods of classical perturbaton theory and recent results by Borichev-Golinskii-Kupin and Favorov-Golinskii, we obtain a quantitative result on the distribution of the discrete spectrum of the operator in the unbounded (outer) component of its Fredholm set.

math.SP↗
arXiv2020

On spectral properties of compact Toeplitz operators on Bergman space with logarithmically decaying symbol and applications to banded matrices

Let $L^2(D)$ be the space of measurable square-summable functions on the unit disk. Let $L^2_a(D)$ be the Bergman space, i.e., the (closed) subspace of analytic functions in $L^2(D)$. $P_+$ stays for the orthogonal projection going from $L^2(D)$ to $L^2_a(D)$. For a function $φ\in L^\infty(D)$, the Toeplitz operator $T_φ: L^2_a(D)\to L^2_a(D)$ is defined as $$ T_φf=P_+φf, \quad f\in L^2_a(D). $$ The main result of this article are spectral asymptotics for singular (or eigen-) values of compact Toeplitz operators with logarithmically decaying symbols, that is $$ φ(z)=φ_1(e^{iθ})\, (1+\log(1/(1-r)))^{-γ},\quad γ>0, $$ where $z=re^{iθ}$ and $φ_1$ is a continuous (or piece-wise continuous) function on the unit circle. The result is applied to the spectral analysis of banded (including Jacobi) matrices.

math.SP↗
arXiv2017

On the instability of the essential spectrum for block Jacobi matrices

We are interested in the phenomenon of the essential spectrum instability for a class of unbounded (block) Jacobi matrices. We give a series of sufficient conditions for the matrices from certain classes to have a discrete spectrum on a half-axis of a real line. An extensive list of examples showing the sharpness of obtained results is provided.

math-ph↗
arXiv2016

A remark on analytic Fredholm alternative

We apply a recent result of Borichev-Golinskii-Kupin on the Blaschke-type conditions for zeros of analytic functions on the complex plane with a cut along the positive semi-axis to the problem of the eigenvalues distribution of the Fredholm-type analytic operator-valued functions.

math.FA↗