Properties of singular integral operators $$varvec{S}_{varvec{alpha ,beta }}$$

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 }).