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

Abstract

For α, β ∈ L ∞ (S 1 ), the singular integral operator S α,β on L 2 (S 1 ) is defined by S α,β f := αP f + 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 ) ⊥ . 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 .