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

Adolf Mirotin

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arXiv · 2512.06552

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

Adolf Mirotin

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Abstract

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.

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BibTeX RIS

Adolf Mirotin. 2026-09-21. On Toeplitz operators on compact Abelian groups and discrete Wiener--Hopf operators. https://arxiv.org/abs/2512.06552

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