Algebras of Generalized Singular Integral Operators with Cauchy kernel

2022/03/09 by Yuanqi Sang, Sang, Yuanqi
Mathematics · #Algebraic and Geometric Analysis #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2203.04615

openalex publication_date 2022/03/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For bounded Lebesgue measurable functions f,g,ϕ and ψ on the unit circle, P₊fP₊+P_-gP₊ +P₊ϕP_-+P_-ψP_- is called a generalized singular integral operator (GSIO) on L²(\mathbbT), 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.

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