Properties of singular integral operators \(S_{\alpha,\beta}\). (English) Zbl 1391.45017
Summary: For \(\alpha , \beta \in L^{\infty } (S^1),\) the singular integral operator \(S_{\alpha ,\beta }\) on \(L^2 (S^1)\) is defined by \(S_{\alpha ,\beta }f:= \alpha Pf+\beta Qf\), where \(P\) denotes the orthogonal projection of \(L^2(S^1)\) onto the Hardy space \(H^2(S^1),\) and \(Q\) denotes the orthogonal projection onto \(H^2(S^1)^{\perp }\). In a recent paper, Nakazi and Yamamoto have studied the normality and self-adjointness of \(S_{\alpha ,\beta }\). This work has shown that \(S_{\alpha ,\beta }\) may have analogous properties to that of the Toeplitz operator. In this paper, we study several other properties of \(S_{\alpha ,\beta }\).
MSC:
| 45P05 | Integral operators |
| 30E20 | Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane |
| 30H10 | Hardy spaces |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 47B20 | Subnormal operators, hyponormal operators, etc. |
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