Properties of singular integral operators $S_{α,β}$

For $α, β\in L^{\infty} (S^1),$ the singular integral operator $S_{α,β}$ on $L^2 (S^1)$ is defined by $S_{α,β}f:= αPf+βQf$, 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 $S_{α,β}.$ This work has shown that $S_{α,β}$ may have analogous properties to that of the Toeplitz operator. In this paper we study several other properties of $S_{α,β}.$

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