Consider the multiplication operator MB in $L^2(\mathbb{T})$, where the symbol B is a finite Blaschke product. In this article, we characterize the commutant of MB 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.

考虑$L^2(\mathbb{T})$中的乘法运算符M B，其中符号B是有限 Blaschke 乘积。在本文中，我们描述了$L^2(\mathbb{T})$中M B的交换子。作为此表征结果的应用，我们明确确定与$M_{z ^2 }$交换的共轭类，或通过在$L^2( \mathbb{T})$。此外，我们在保持整个 Hardy 空间、模型空间和 Beurling 型子空间不变的情况下分析了它们的性质。此外，我们扩展了关于有限 Blaschke 乘积情况下共轭的研究。