Algebras of generalized Cauchy singular integral operators. (English) Zbl 1539.47122
Summary: For bounded Lebesgue measurable functions \(f, g, \phi, \psi\) on the unit circle, \(P_+fP_++P_-gP_+ +P_+\phi P_-+P_-\psi P_-\) is called a generalized Cauchy singular integral operator on \(L^2(\mathbb{T})\), where \(P_+\) is the Riesz projection, \(P_-=I_{L^2}-P_+\). In this paper, we relate generalized Cauchy singular integral operators to a number of operators, including Cauchy singular integral operators, (dual) truncated Toeplitz operators, the dilation of truncated Toeplitz operators, Foguel-Hankel operators, \(2\times 2\) block Toeplitz operators, Toeplitz plus Hankel operators, etc. We obtain the characteristic equation of generalized Cauchy singular integral operators, and establish the short exact sequences associated of the \(C^\ast\)-algebras generated by generalized Cauchy singular integral operators with bounded or quasi-continuous symbols. As a consequence, we discuss the spectral inclusion theorem and calculate its Fredholm index. Moreover, we give the necessary and sufficient conditions for invertibility (Fredholmness) of generalized Cauchy singular integral operators via Wiener-Hopf factorization.
MSC:
| 47L80 | Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.) |
| 47G10 | Integral operators |
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
| 47C10 | Linear operators in \({}^*\)-algebras |
| 47A68 | Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators |
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