Lie Algebras of Bounded Operators

by Beltija, Daniel; Sabac, Mihai

• ISBN13: 9783764364045
• ISBN10: 3764364041
• Format: Hardcover
• Copyright: 2000-11-01
• Publisher: Birkhauser

Summary

There is a fruitful and fascinating interaction between infinite dimensional operator theory (particularly decomposable, scalar and spectral generalized operator theory due to C. Foias and I. Colojoara) and Lie algebra theory. The present book is the first devoted to this field, ranging from some short historical notes to the most recent developments. Nilpotence criteria, infinite dimensional variants of Lie's theorem for solvable systems of bounded operators, spectral properties of elements of semisimple Lie algebras and simultaneous triangularisation are expounded. The book is self-contained and features an extensive bibliography. It is aimed at postgraduate students and researchers who are introduced to an interesting recent area of research and will learn some new methods useful for both of the domains - operator theory and Lie algebra theory.

Table of Contents

Introduction	vii
I Preliminaries
Lie Algebras
Basic facts	1	(3)
Ideals, solvability, nilpotence, radical and semisimplicity	4	(4)
Sp classes of solvable Lie algebras	8	(2)
Radical splitting theorems	10	(1)
Cartan subalgebras	11	(4)
The Killing form and compact Lie algebras	15	(1)
Lie *-algebras	16	(7)
Complexes
Generalities	23	(8)
Banach space complexes	31	(2)
Koszul complexes	33	(19)
Operations with the Koszul complexes: duality and tensor products	52	(11)
Spectral Theory in Complex Banach Space
General spectral theory and decomposable operators	63	(5)
The transformation of the spectral maximal subspaces by bounded operators	68	(3)
Special classes of decomposable operators	71	(10)
Notes	77	(4)
II The Commutators and Nilpotence Criteria
An asymptotic formula for the commutators	81	(8)
Nilpotence criteria in an associative algebra	89	(2)
Quasinilpotence and nilpotence criteria in complex Banach algebras	91	(4)
Nilpotent elements in LM-decomposable Lie subalgebras of an associative algebra	95	(6)
Nilpotent elements in LM-decomposable Lie algebras of bounded linear operators	101	(2)
Notes	101	(2)
III Infinite-dimensional Variants of the Lie and Engel Theorems
Weights for operator algebras	103	(11)
Invariant subspaces for LM-decomposable Lie algebras of bounded operators	114	(7)
The irreducible representations of an LM-decomposable Lie algebra. Infinite-dimensional variant of Lie's Theorem on a complex Banach space	121	(5)
The associative envelope of a Lie algebra of quasinilpotent operators	126	(3)
Commutativity modulo the Jacobson radical	129	(4)
Notes	131	(2)
IV Spectral Theory for Solvable Lie Algebras of Operators
Spectral theory for representations of Lie algebras	133	(17)
Spectral theory for systems of operators generating nilpotent Lie algebras	150	(13)
The Cartan-Taylor spectrum of a locally solvable Lie algebra of operators	163	(10)
Lie ideals of generalized spectral operators	173	(8)
Notes	180	(1)
V Semisimple Lie Algebras of Operators
Lie subalgebra with involution consisting of bounded operators on a complex Banach space. Normal elements given by a space of self-adjoint operators	181	(7)
Individual spectral properties in ideally finite semisimple Lie algebras of operators	188	(10)
Semisimple Lie algebras of compact quasinilpotent operators	198	(5)
Notes	202	(1)
Bibliography	203	(12)
Index	215	(4)
List of Symbols	219