Algebras of Generalized Singular Integral Operators with Cauchy kernel

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.