Commutants and complex symmetry of finite Blaschke product multiplication operator in $L^2(\mathbb{T})$.

Consider the multiplication operator M_B in $L^2(\mathbb{T})$ , where the symbol B is a finite Blaschke product. In this article, we characterize the commutant of M_B in $L^2(\mathbb{T})$. As an application of this characterization result, we explicitly determine the class of conjugations commuting with $M_{z^2}$ or making $M_{z^2}$ complex symmetric by introducing a new class of conjugations in $L^2(\mathbb{T})$. Moreover, we analyse their properties while keeping the whole Hardy space, model space and Beurling-type subspaces invariant. Furthermore, we extended our study concerning conjugations in the case of finite Blaschke products. [ABSTRACT FROM AUTHOR]