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

TL;DR

This paper characterizes the commutant of multiplication operators with finite Blaschke product symbols in L^2(T), explores complex symmetry via conjugations, and extends the analysis to various subspaces and specific cases.

Contribution

It provides a detailed characterization of the commutant of M_B in L^2(T) and introduces new conjugations to analyze complex symmetry, extending previous results.

Findings

Explicit description of the commutant of M_B in L^2(T)

Identification of conjugations commuting with M_{z^2}

Extension of conjugation analysis to finite Blaschke products

Abstract

Consider the multiplication operator $M_{B}$ in $L^2(\T)$, where the symbol $B$ is a finite Blaschke product. In this article, we characterize the commutant of $M_{B}$ in $L^2(\T)$, noting the fact that $L^2(\T)$ is not an RKHS. 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 $L^2(\T)$. Moreover, we analyze 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.