Simultaneous triangularization
type: Item
instance of: scholarly article
title: Simultaneous triangularization
zbMATH Open document ID: 0981.15007
DOI: 10.1007/978-1-4612-1200-3
author: P. Rosenthal
author: H. Radjavi
published in: Universitext
publication date: 2000-02-22T00:00:00Z
review text: Simultaneous triangularization of operators can be regarded as a kind of generalized commutativity: it implies (and is implied by) certain consequences of commutativity. It a priori requires the existence of common invariant subspaces [concerning this problem, the authors' book ``Invariant subspaces'' (1973; Zbl 0269.47003) became famous], and thus it particularly applies to operators that are closely related to compact operators. Of course, the concept originates from finite dimension. The first five chapters of the book are devoted to this classical development, with much modern insight. The sixth chapter presents basic facts on compact operators (including Hilden's proof of Lomonosov's theorem), and the next two chapters show that most of the finite dimensional results have satisfactory extensions to compact operators. On the other hand, the last ninth chapter provides a number of counterexamples in the case of general infinite-dimensional operators. The authors have done a great job in finding a simple presentation of a beautiful theory, and collecting all relevant references. Their own results and simplifications represent a substantial part of the book. Each chapter ends with corresponding historical remarks. One can find even the most recent results, in particular, Turovskiĭ's theorem: A semigroup of compact quasi-nilpotent operators on a Banach space is triangularizable. Nevertheless, this interesting development does not finish with the book. For instance, the authors have recently found even a simpler proof of Theorem 2.3.10 in their preprint ``On commutators of idempotents'' (2000). Other papers related to the book have appeared, for instance: \textit{R. Drnovšek} [Hyperinvariant subspaces for operator semigroups with commutators of rank at most one, Houston J. Math. 26, No. 3, 543-548 (2000; Zbl 0981.47005)]; \textit{R. Yahagi} [On simultaneous triangularization of commutants, Acta Sci. Math. 66, No. 3-4, 711-718 (2000; Zbl 1062.47504)]; \textit{G. MacDo…
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15-02
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 47A65
Mathematics Subject Classification ID: 47C05
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47-02
zbMATH DE Number: 1404753
zbMATH Keywords: semigroup of operators
zbMATH Keywords: idempotent
zbMATH Keywords: nilpotent
zbMATH Keywords: simultaneous triangularization
zbMATH Keywords: generalized commutativity
zbMATH Keywords: invariant subspaces
zbMATH Keywords: compact operators
zbMATH Keywords: spectrum
zbMATH Keywords: spectral radius
zbMATH Keywords: trace
zbMATH Keywords: commutators
zbMATH Keywords: Banach algebras
Wikidata QID: Q56391051
reviewed by: Jaroslav Zemánek
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-1-4612-1200-3
OpenAlex ID: W4213199622
description: scientific article; zbMATH DE number 1404753