On the similarity of restriction of the operator to an invariant subspace

TL;DR

This paper investigates when two restricted multiplication operators on the Bergman space are similar, providing sufficient conditions for their similarity in a general setting.

Contribution

It offers new sufficient conditions for the similarity of restriction operators of the Bergman space multiplication operator, addressing a question posed by K. Zhu.

Findings

Identifies conditions under which restriction operators are similar

Provides a partial answer to K. Zhu's question

Extends understanding of operator similarity on invariant subspaces

Abstract

Let $M_{z}$ be the multiplication operator on the Bergman space and $M_{I}$ denote the restriction of $M_{z}$ to an invariant subspace $I$. A question raised by K. Zhu is that when are two restriction operators $M_{I}$ and $M_{J}$ are similar? In this note, we give some sufficient conditions of this problem in a general case.