Algebras of Generalized Singular Integral Operators with Cauchy kernel

TL;DR

This paper studies a broad class of generalized singular integral operators on the unit circle, establishing their algebraic structure, spectral properties, and invertibility criteria using advanced operator theory techniques.

Contribution

It introduces the algebraic framework for GSIOs, relates them to known operators, and provides spectral and invertibility conditions via Winer-Hopf factorization.

Findings

Established short exact sequences for C*-algebras generated by GSIOs.

Computed spectra and Fredholm indices of GSIOs.

Provided necessary and sufficient conditions for invertibility and Fredholmness.

Abstract

For bounded Lebesgue measurable functions $f,g,\phi$ and $\psi$ on the unit circle, $P_{+}fP_{+}+P_{-}gP_{+} +P_{+}\phi P_{-}+P_{-}\psi P_{-}$ is called a generalized singular integral operator (GSIO) on $L^{2}(\mathbb{T})$, where $P_{+}$ is the Riesz projection, $P_{-}=I-P_{+}.$ In this paper, we relate GSIOs to a number of operators, including Cauchy singular integral operator, (dual) truncated Toeplitz operator, Foguel-Hankel operator, multiplication operator, Toeplitz plus Hankel operator etc. We establish the short exact sequences associated of the $C^{*}-$algebras generated by GSIOs with bounded or quasi-continuous symbols. As a consequence we obtain the spectra of various classes of GSIOs, the spectral inclusion theorem and comput the Fredholm index of GSIOs. Moreover, we gave the necessary and sufficient conditions for invertibility(Fredholmness) of GSIOs via Winer-Hopf factorization.