Properties of singular integral operators $S_{\alpha ,\beta }$

For $\alpha ,\beta \in L^{\mathrm{\infty }}(S^1),$ the singular integral operator $S_{\alpha ,\beta }$ on $L^2(S^1)$ is defined by $S_{\alpha,\beta}f:= \alpha Pf+\beta Qf$,where$P$denotestheorthogonalprojectionof$L^2(S^1)$ onto the Hardy space $H^2(S^1),$ and $Q$ denotes the orthogonal projection onto $H^2(S^1{)}^{\perp }\mathrm{.}$ In a recent paper Nakazi and Yamamoto have studied the normality and self-adjointness of $S_{\alpha ,\beta }\mathrm{.}$ 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 $S_{\alpha ,\beta }\mathrm{.}$