Simultaneous Triangularization

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.

This volume deals both with the finite-dimensional and the compact theorems about triangularization. It is designed to appeal to two different audiences. The first four chapters should be of interest to linear-algebraists and other algebraists (e.g. people interested in finite-dimensional representations) who use algebras or semigroups of matrices. The remaining chapters concerned with the structures of linear operators on Banach spaces will attract operator theorists.

One: Algebras of Matrices.- 1.1 The Triangularization Lemma.- 1.2 Burnside's Theorem.- 1.3 Triangularizability of Algebras of Matrices.- 1.4 Triangularization and the Radical.- 1.5 Block Triangularization and Characterizations of Triangularizability.- 1.6 Approximate Commutativity.- 1.7 Nonassociative Algebras.- 1.8 Notes and Remarks.- Two: Semigroups of Matrices.- 2.1 Basic Definitions and Propositions.- 2.2 Permutable Trace.- 2.3 Zero-One Spectra.- 2.4 Notes and Remarks.- Three: Spectral Conditions on Semigroups.- 3.1 Reduction to the Field of Complex Numbers.- 3.2 Permutable Spectrum.- 3.3 Submultiplicative Spectrum.- 3.4 Conditions on Spectral Radius.- 3.5 The Dominance Condition on Spectra.- 3.6 Notes and Remarks.- Four: Finiteness Lemmas and Further Spectral Conditions.- 4.1 Reductions to Finite Semigroups.- 4.2 Subadditive and Sublinear Spectra.- 4.3 Further Multiplicative Conditions on Spectra.- 4.4 Polynomial Conditions on Spectra.- 4.5 Notes and Remarks.- Five: Semigroups of Nonnegative Matrices.- 5.1 Decomposability.- 5.2 Indecomposable Semigroups.- 5.3 Connections with Reducibility.- 5.4 Notes and Remarks.- Six: Compact Operators and Invariant Subspaces.- 6.1 Operators on Banach Spaces.- 6.2 Compact Operators.- 6.3 Invariant Subspaces for Compact Operators.- 6.4 The Riesz Decomposition of Compact Operators.- 6.5 Trace-Class Operators on Hilbert Space.- 6.6 Notes and Remarks.- Seven: Algebras of Compact Operators.- 7.1 The Definition of Triangularizability.- 7.2 Spectra from Triangular Forms.- 7.3 Lomonosov's Lemma and McCoy's Theorem.- 7.4 Transitive Algebras.- 7.5 Block Triangularization and Applications.- 7.6 Approximate Commutativity.- 7.7 Notes and Remarks.- Eight: Semigroups of Compact Operators.- 8.1 Quasinilpotent Compact Operators.- 8.2 AGeneral Approach.- 8.3 Permutability and Submultiplicativity of Spectra.- 8.4 Subadditivity and Sublinearity of Spectra.- 8.5 Polynomial Conditions on Spectra.- 8.6 Conditions on Spectral Radius and Trace.- 8.7 Nonnegative Operators.- 8.8 Notes and Remarks.- Nine: Bounded Operators.- 9.1 Collections of Nilpotent Operators.- 9.2 Commutators of Rank One.- 9.3 Bands.- 9.4 Nonnegative Operators.- 9.5 Notes and Remarks.- References.- Notation Index.- Author Index.