Brown-Halmos type theorems for generalized Cauchy singular integral operators and applications

We investigate the commutativity and semi-commutativity of generalized singular integral operators of the form $P_{+} f P_{+} + P_{-} g P_{+} + P_{+} u P_{-} + P_{-} v P_{-}$ on $L^{2}$, where $P_{+}$ denotes the Riesz projection and $P_{-}=I-P_{+}$. Building on this analysis, we develop a unified approach to studying the algebraic properties of operator classes on $L^{2}$ generated by multiplication operators together with the Riesz projection. These classes include, but are not limited to, Toeplitz+Hankel operators, singular integral operators, Foguel--Hankel operators, and asymmetric dual truncated Toeplitz operators. We provide complete characterizations of (i) the quasinormality of singular integral operators, and (ii) the necessary and sufficient conditions under which the product of two asymmetric dual truncated Toeplitz operators is again an asymmetric dual truncated Toeplitz operator. In addition, our methods provide new proofs of several known results, including the classical Brown-Halmos theorems and the commutativity of Hankel operators, singular integral operators, and dual truncated Toeplitz operators. We also improve the conditions for the normality of singular integral operators.