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Consider the multiplication operator $M_{B}$ in $L^2(\T)$, where the symbol$B$ is a finite Blaschke product. In this article, we characterize thecommutant of $M_{B}$ in $L^2(\T)$, noting the fact that $L^2(\T)$ is not anRKHS. As an application of this characterization result, we explicitlydetermine the class of conjugations commuting with $M_{z^2}$ or making$M_{z^2}$ complex symmetric by introducing a new class of conjugations in$L^2(\T)$. Moreover, we analyze their properties while keeping the whole Hardyspace, model space, and Beurling-type subspaces invariant. Furthermore, weextended our study concerning conjugations in the case of finite Blaschke.

Commutants and Complex Symmetry of Finite Blaschke Product Multiplication Operator in $\bm{L^2(\T)}$