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

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 $S_{\alpha,\beta}.$