arXiv subjects

Adolf Mirotin

Publications and source records attributed to Adolf Mirotin.

6 recordsLinked to original sources

arXiv2026

On Toeplitz operators on compact Abelian groups and discrete Wiener--Hopf operators

This paper introduces the concept of a rotation number for a continuous, non-degenerate two-dimensional vector field (a zero-free complex-valued function) on a compact connected Abelian group. This concept generalizes the notion of a finite rotation number for such groups, previously introduced by the author. Using this concept, a Gohberg-Krein index formula is derived for semi-Fredholm Toeplitz operators with continuous symbols defined on such groups. Criteria for these operators to be semi-Fredholm are established, and their essential spectra are described. As a by-product for the continuous symbol case, conditions for Fredholmness and semi-Fredholmness are established, and the Fredholm index of Wiener-Hopf operators over a linearly ordered discrete Abelian group is calculated in terms of their symbols. Spectral properties-including the spectra and essential spectra-of the Wiener-Hopf operators under consideration are also described.

math.FA↗

arXiv2026

Functions of continuous Cesáro operators

We describe holomorphic functions and fractional powers of Cesáro operators in $L^2(\mathbb{R})$, $L^2(\mathbb{R}_+)$, and $L^2[0,1]$. Logarithms of Cesáro operators are introduced as well and their spectral properties are studied. Several examples are considered.

math.FA↗

arXiv2025

On a unique two-dimensional integral operator homogeneous with respect to all orientation preserving linear transformations

In this paper, we consider a two-dimensional operator with an antisymmetric integral kernel, recently introduced by Z. Avetisyan and A. Karapetyants in connection to the study of general homogeneous operators. This is the unique two-dimensional operator that has an antisymmetric kernel homogeneous with respect to all orientation-preserving linear transformations of the plane. It is shown that the operator under consideration interacts naturally, both in Cartesian and polar coordinates, with projective tensor products of some classical functional spaces, such as Lebesgue, Hardy, and Hölder spaces; conditions for their boundedness as operators acting from these spaces to Banach lattices of measurable functions and estimates of their norms are given.

math.FA↗

arXiv2023

Some remarks on the solution of the cell growth equation

The analytical solution to the initial-boundary value problem for the cell growth equation was given in the paper Zaidi A. A., Van Brunt B., Wake G.C., Solutions to an advanced functional partial differential equation of the pantograph type, Proc. R. Soc. A 471: 20140947 (2015). In this note, we simplify the arguments given in the paper mentioned above by using the theory of operator semigroups.

math.AP↗

arXiv2020

Boundedness of Hausdorff operators on Hardy spaces $H^1$ over locally compact groups

Results of Liflyand and collaborators on the boundedness of Hausdorff operators on the Hardy space $H^1$ over finite-dimensional real space generalized to the case of locally compact groups that are spaces of homogeneous type. Special cases and examples of compact Lie groups, homogeneous groups (in particular the Heisenberg group) and finite-dimensional spaces over division rings are considered.

math.FA↗

arXiv2019

Perturbation determinants on Banach spaces and operator differentiability for Hirsch functional calculus

We consider a perturbation determinant for pairs of nonpositive (in a sense of Komatsu) operators on Banach space with nuclear difference and prove a generalization of the important formula for the logarithmic derivative of this determinant. To this end the Frechet differentiability of operator monotonic (negative complete Bernstein) functions of negative and nonpositive operators on Banach spaces is investigated. The results may be regarded as a contribution to the Hirsch functional calculus.

math.FA↗