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On the Hilbert space \(\widetilde {L}_{2}(\mathbb {T})\) the singular integral operator with two shifts and conjugation \(K=P_{+}+\left [aI+\left (\sum \limits _{j=0}^{m} a_{j}U_{\alpha }^{j}\right )U_{\beta }C\right ]P_{-}\) is considered, where \(P_{\pm }\) are the Cauchy projectors, \(a,a_{j}\) , \(j=\overline {0,m}\) , are continuous functions on the unit circle \(\mathbb {T}\) , \(U_{\alpha }\) and \(U_{\beta }\) are non-Carleman and Carleman shift operators, respectively, both preserving the orientation on \(\mathbb {T}\) , and C is the operator of complex conjugation. An estimate for the dimension of the kernel of the operator K is obtained. We also consider the operator \(M=[aI+(a_{0}I+a_{1}U_{\alpha })U_{\gamma }C]P_{+}+P_{-}\) , where \(U_{\gamma }\) is a Carleman shift operator changing the orientation on \(\mathbb {T}\) .

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On a Singular Integral Operator with Two Shifts and Conjugation

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  • 发表日期：

    2025年

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  • Rui C. Marreiros

摘要

On the Hilbert space \(\widetilde {L}_{2}(\mathbb {T})\) the singular integral operator with two shifts and conjugation \(K=P_{+}+\left [aI+\left (\sum \limits _{j=0}^{m} a_{j}U_{\alpha }^{j}\right )U_{\beta }C\right ]P_{-}\) is considered, where \(P_{\pm }\) are the Cauchy projectors, \(a,a_{j}\) , \(j=\overline {0,m}\) , are continuous functions on the unit circle \(\mathbb {T}\) , \(U_{\alpha }\) and \(U_{\beta }\) are non-Carleman and Carleman shift operators, respectively, both preserving the orientation on \(\mathbb {T}\) , and C is the operator of complex conjugation. An estimate for the dimension of the kernel of the operator K is obtained. We also consider the operator \(M=[aI+(a_{0}I+a_{1}U_{\alpha })U_{\gamma }C]P_{+}+P_{-}\) , where \(U_{\gamma }\) is a Carleman shift operator changing the orientation on \(\mathbb {T}\) .