# Properties of singular integral operators \(\varvec{S}_{\varvec{\alpha ,\beta }}\)

## Abstract

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 \(S_{\alpha ,\beta }\).

## Keywords

Singular integral operator Toeplitz operator Hardy space## 2010 Mathematics Subject Classification

45E10 47B35 47B20 30D55## Notes

### Acknowledgements

The authors would like to thank the referee for his/her important remarks which led to an improved article. The first author is supported by the NBHM Postdoctoral Fellowship, Government of India. The second author is supported by the Feinberg Postdoctoral Fellowship of the Weizmann Institute of Science.

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