Normal Singular Integral Operators with Cauchy Kernel on L 2

Let α and β be functions in ${L^infty(mathbb{T})}$ , where ${mathbb{T}}$ is the unit circle. Let P denote the orthogonal projection from ${L^2(mathbb{T})}$ onto the Hardy space ${H^2(mathbb{T})}$ , and Q = I ? P, where I is the identity operator on ${L^2(mathbb{T})}$ . This paper is concerned with the singular integral operators S _α,β on ${L^2(mathbb{T})}$ of the form S _α,β f = αPf + βQf, for ${f in L^2(mathbb{T})}$ . In this paper, we study the normality of S _α,β which is related to the Brown–Halmos theorem for the normal Toeplitz operator on ${H^2(mathbb{T})}$ .