Simultaneous Triangularization

by Radjavi, Heydar; Rosenthal, Peter

• ISBN13: 9780387984667
• ISBN10: 0387984666
• Format: Paperback
• Copyright: 2000-03-01
• Publisher: Springer Verlag

Summary

A collection of matrices is said to be triangularizable if there is an invertible matrix S such that S1 AS is upper triangular for every A in the collection. This generalization of commutativity is the subject of many classical theorems due to Engel, Kolchin, Kaplansky, McCoy and others. The concept has been extended to collections of bounded linear operators on Banach spaces: such a collection is defined to be triangularizable if there is a maximal chain of subspaces of the Banach space, each of which is invariant under every member of the collection. Most of the classical results have been generalized to compact operators, and there are also recent theorems in the finite-dimensional case. This book is the first comprehensive treatment of triangularizability in both the finite and infinite-dimensional cases. It contains numerous very recent results and new proofs of many of the classical theorems. It provides a thorough background for research in both the linear-algebraic and operator-theoretic aspects of triangularizability and related areas. More generally, the book will be useful to anyone interested in matrices or operators, as many of the results are linked to other topics such as spectral mapping theorems, properties of spectral radii and traces, and the structure of semigroups and algebras of operators. It is essentially self-contained modulo solid courses in linear algebra (for the first half) and functional analysis (for the second half), and is therefore suitable as a text or reference for a graduate course. ^

Table of Contents

Preface	vii
Algebras of Matrices	1	(26)
The Triangularization Lemma	1	(3)
Burnside's Theorem	4	(2)
Triangularizability of Algebras of Matrices	6	(4)
Triangularization and the Radical	10	(2)
Block Triangularization and Characterizations of Triangularizability	12	(5)
Approximate Commutativity	17	(4)
Nonassociative Algebras	21	(4)
Notes and Remarks	25	(2)
Semigroups of Matrices	27	(16)
Basic Definitions and Propositions	27	(6)
Permutable Trace	33	(3)
Zero-One Spectra	36	(5)
Notes and Remarks	41	(2)
Spectral Conditions on Semigroups	43	(32)
Reduction to the Field of Complex Numbers	43	(8)
Permutable Spectrum	51	(4)
Submultiplicative Spectrum	55	(7)
Conditions on Spectral Radius	62	(8)
The Dominance Condition on Spectra	70	(3)
Notes and Remarks	73	(2)
Finiteness Lemmas and Further Spectral Conditions	75	(29)
Reductions to Finite Semigroups	75	(5)
Subadditive and Sublinear Spectra	80	(9)
Further Multiplicative Conditions on Spectra	89	(4)
Polynomial Conditions on Spectra	93	(9)
Notes and Remarks	102	(2)
Semigroups of Nonnegative Matrices	104	(26)
Decompossability	104	(11)
Indecomposable Semigroups	115	(12)
Connections with Reducibility	127	(2)
Notes and Remarks	129	(1)
Compact Operators and Invariant Subspaces	130	(21)
Operators on Banach Spaces	130	(3)
Compact Operators	133	(2)
Invariant Subspaces for Compact Operators	135	(3)
The Riesz Decomposition of Compact Operators	138	(4)
Trace-Class Operators on Hilbert Space	142	(7)
Notes and Remarks	149	(2)
Algebras of Compact Operators	151	(42)
The Definition of Triangularizability	151	(4)
Spectra from Triangular Forms	155	(7)
Lomonosov's Lemma and McCoy's Theorem	162	(5)
Transitive Algebras	167	(6)
Block Triangularization and Applications	173	(10)
Approximate Commutativity	183	(6)
Notes and Remarks	189	(4)
Semigroups of Compact Operators	193	(51)
Quasinilpotent Compact Operators	193	(6)
A General Approach	199	(6)
Permutability and Submultiplicativity of Spectra	205	(4)
Subadditivity and Sublinearity of Spectra	209	(4)
Polynomial Conditions on Spectra	213	(3)
Conditions on Spectral Radius and Trace	216	(7)
Nonnegative Operators	223	(18)
Notes and Remarks	241	(3)
Bounded Operators	244	(40)
Collections of Nilpotent Operators	244	(6)
Commutators of Rank One	250	(12)
Bands	262	(10)
Nonnegative Operators	272	(9)
Notes and Remarks	281	(3)
References	284	(23)
Notation Index	307	(2)
Author Index	309	(6)
Subject Index	315