论文标题
cauchy内核的广义单数积分运算符的代数
Algebras of Generalized Singular Integral Operators with Cauchy kernel
论文作者
论文摘要
对于有限的lebesgue可测量函数$ f，g，ϕ $和$ψ$在单位圈子上，$ p _ {+} fp _ {+}+p _ { - } gp _ {+}+}+p _ {+p _ {+}+} (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运算符，乘法运算符，Toeplitz Plus Hankel Operator等。我们建立了由GSIOS生成的$ C^{*} - $ c^{*}的简短序列，该序列由gsios生成的$ c^{*} - 带有有限或准连续符号。结果，我们获得了各种GSIO的光谱，频谱包含定理并计算GSIOS的Fredholm指数。此外，我们通过Winer-HOPF分解了GSIO的可逆性（Fredholmness）提供了必要的条件。
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 $L^{2}(\mathbb{T})$, 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.
