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.

Inhalt
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.