On the similarity of powers of operators with flag structure

2023/12/27 by Jianming Yang, Yang, Jianming, Kui Ji +1
Mathematics · #Advanced Topics in Algebra #Algebraic and Geometric Analysis #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2312.16459

openalex publication_date 2023/12/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let L²_a(\mathbbD) be the classical Bergman space and denote M_h for the multiplication operator by a function h. Let B be a finite Blaschke product with order n.An open question proposed by R. G. Douglas is whether the operators M_B on L²_a(\mathbbD) similar to ⊕₁ⁿ M_z on ⊕₁ⁿ L²_a(\mathbbD)? The question was answered in the affirmative, not only for Bergman space but also for many other Hilbert spaces with reproducing kernels.Since the operator M_z^* is in Cowen-Douglas class B₁(\mathbbD) in many cases, Douglas's question can be expressed as a version for operators in B₁(\mathbbD), and it is affirmative for many operators in B₁(\mathbbD).A natural question is how about Douglas's question in the version for operators in Cowen-Douglas class B_n(\mathbbD) (n>1)? In this paper, we investigate a family of operators, which are in a norm dense subclass of Cowen-Douglas class B₂(\mathbbD), and give a negative answer.This indicates that Douglas's question cannot be directly generalized to general Hilbert spaces with vector-valued analytical reproducing kernel.

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