Simultaneous TriangularizationSpringer, 19.11.2012 - 319 Seiten 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. |
Inhalt
1 | |
Semigroups of Matrices | 27 |
Spectral Conditions on Semigroups | 43 |
Finiteness Lemmas | 75 |
Semigroups of Nonnegative Matrices | 104 |
Compact Operators | 130 |
Algebras of Compact Operators | 151 |
Semigroups of Compact Operators | 193 |
Bounded Operators | 244 |
References | 284 |
306 | |
315 | |
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Häufige Begriffe und Wortgruppen
algebra assume Banach space band block triangularization bounded Burnside's Theorem chain of invariant column commutative compact operators contains contradiction Corollary decomposable defined Definition denote diagonal dimension direct sum easily seen eigenvalues entries equivalent finite finite-dimensional finite-rank operators follows from Lemma Hence Hilbert space hypothesis idempotent of rank implies inherited by quotients integer invertible irreducible semigroup kernel linear span linear transformations LP(X M/M_ M₁ minimal idempotent multiplicity need only show nilpotent nilpotent operators nonnegative matrices nontrivial invariant subspace norm Note pair permutable permutation matrix Perron-Frobenius Theorem polynomial positive prove quasinilpotent operators Radjavi Radjavi-Rosenthal range reducible right ideal S₁ satisfying scalar semigroup of compact semigroup of nonnegative sequence span spectral radius spectrum subalgebra sublinear submultiplicative subset suffices to show trace trace-class triangularizable triangularizing chain unitary vector zero