Almost commuting matrices and a quantitative version of the Brown- Douglas-Fillmore theorem
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.
scientific article; zbMATH DE number 4208866
scientific article