Almost commuting matrices and a quantitative version of the Brown- Douglas-Fillmore theorem
type: Item
instance of: scholarly article
title: Almost commuting matrices and a quantitative version of the Brown- Douglas-Fillmore theorem
zbMATH Open document ID: 0731.47009
DOI: 10.1007/BF02398885
author: I. David Berg
author: Kenneth R. Davidson
published in: Acta Mathematica
publication date: 1991-01-01T00:00:00Z
review text: The authors give a constructive proof of a theorem of \textit{L. G. Brown}, \textit{R. G. Douglas} and \textit{P. A. Fillmore} [Lect. Notes Math. 345, 58- 128 (1973; Zbl 0277.46053)] which yields a quantitative version subject to a certain natural resolvent condition. The most important special case in their proof is the annulus \(A=\{\lambda \in {\mathbb{C}}:\) \(R_ 1\leq | \lambda | \leq R_ 2\}.\) Theorem. Let T be an operator on a Hilbert space with \(\sigma_ e(T)=A=\{\lambda \in {\mathbb{C}}:\) \(R_ 1\leq | \lambda | \leq R_ 2\}\). Suppose \(\| T\| =R_ 2\) and \(\| T^{-1}\| =R_ 1^{-1}\). Then there is an operator K such that \(\| K\| \leq 104\| T^*T-TT^*\|^{1/2}\) and such that T-K is normal with spectrum A. If T is essentially normal, then K may be taken to be compact. The theorem above is applied to the proof of the quantitative version of the theorem of BDF. The authors are interested in showing that in the case of planar sets, the results of BDF can be explained in a purely operator theoretic way. Moreover this approach is very good for one to understand the compact perturbation. In the last section eight problems are given.
reviewed by: Takahiko Nakazi
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 47B15
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47A66
Mathematics Subject Classification ID: 47A20
zbMATH DE Number: 4208866
zbMATH Keywords: Brown-Douglas-Fillmore theorem
zbMATH Keywords: quantitative version
zbMATH Keywords: resolvent condition
zbMATH Keywords: essentially normal
zbMATH Keywords: compact perturbation
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02398885
OpenAlex ID: W2073801900
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description: scientific article; zbMATH DE number 4208866