Similarity to the backward shift operator on the Dirichlet space
type: Item
instance of: scholarly article
title: Similarity to the backward shift operator on the Dirichlet space
zbMATH Open document ID: 1399.47071
author: Hyun-Kyoung Kwon
published in: Journal of Operator Theory
publication date: 2016-12-01T00:00:00Z
Mathematics Subject Classification ID: 47A99
Mathematics Subject Classification ID: 47B32
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 30J99
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 6658796
zbMATH Keywords: backward shift
zbMATH Keywords: Cowen-Douglas operator
zbMATH Keywords: Dirichlet space
zbMATH Keywords: eigenvector bundle
zbMATH Keywords: reproducing kernel
zbMATH Keywords: similarity
MaRDI profile type: Publication
full work available at URL: jot.2015sep19.2079
OpenAlex ID: W2526226797
DOI: 10.7900/JOT.2015SEP19.2079
review text: In this paper, the author gives a necessary condition for a Cowen-Douglas operator to be similar to the backward shift operator \(D_{n}^{\ast}\) on the Dirichlet space. The main theorem is the following (Theorem 2.2). Let \(H\) be a Hilbert space. Let \(D\) the unit disk. Suppose \(T:H\rightarrow H\) is a bounded linear operator that satisfies the following four properties: (1) \(\sum_{k=0}^\infty \frac{\| T^k\|^2}{k+1}\leq 1\). (2) The closed linear span of \(\bigcup_{\lambda \in D}\ker (T-\lambda I)=H\). (3) The subspaces \(\ker(T-\lambda I)\) analytically depend on \(\lambda\in D\). (4) There is an \(n\in N\) such that \(\dim(\ker(T-\lambda I))=n\) for all \(\lambda\in D\). If \(T\) is similar to the backward shift operator \(D_{n}^{\ast}\), then for some bounded subharmonic function \(\phi :D\rightarrow C\) we have NEWLINE\[NEWLINE\Delta \phi (\lambda)\geq \left \| \frac{\partial \Pi (\lambda)}{\partial \lambda}\right \|_{G_{2}}^{2}+\ell (\lambda)\quad \text{ for all \(\lambda \in D\),}NEWLINE\]NEWLINE where \(\Delta =\frac{1}{4}t(\frac{\partial ^{2}}{\partial x^{2}}+\frac{\partial^ {2}}{\partial y^{2}})\) is the normalized Laplacian, \(\Pi (\lambda):H \rightarrow \ker (T-\lambda I)\) is the orthogonal projection onto \(\ker (T-\lambda I)\), \(\| \cdot \|_{G_{2}}^{2}\) denotes the Hilbert-Schmidt operator norm, and NEWLINE\[NEWLINE l(\lambda)=\begin{cases} \frac{n[\log (1-|\lambda |^{2})+|\lambda |^{2}]} {[\log (1-|\lambda |^{2})(1-|\lambda |^{2})]^{2}}&\text{if } | \lambda |>0, \\ -\frac{n}{2} & \text{if }\lambda =0.\end{cases} NEWLINE\]
reviewed by: Joaquim Martin
description: scientific article; zbMATH DE number 6658796