Operator Theory

I Preliminaries.- A Lie Algebras.- x 1 Basic facts.- x 2 Ideals, solvability, nilpotence, radical and semisimplicity.- x 3 SP classes of solvable Lie algebras.- x 4 Radical splitting theorems.- x 5 Cartan subalgebras.- x 6 The Killing form and compact Lie algebras.- x 7 Lie *-algebras.- B Complexes.- x 8 Generalities.- x 9 Banach space complexes.- x 10 Koszul complexes.- x 11 Operations with the Koszul complexes: duality and tensor products.- C Spectral Theory in Complex Banach Space.- x 12 General spectral theory and decomposable operators.- x 13 The transformation of the spectral maximal subspaces by bounded operators.- x 14 Special classes of decomposable operators.- Notes.- II The Commutators and Nilpotence Criteria.- x 15 An asymptotic formula for the commutators.- x 16 Nilpotence criteria in an associative algebra.- x 17 Quasinilpotence and nilpotence criteria in complex Banach algebras.- x 18 Nilpotent elements in LM-decomposable Lie subalgebras of an associative algebra.- x 19 Nilpotent elements in LM-decomposable Lie algebras of bounded linear operators.- Notes.- III Infinite-dimensional Variants of the Lie and Engel Theorems.- x 20 Weights for operator algebras.- x 21 Invariant subspaces for LM-decomposable Lie algebras of bounded operators.- x 22 The irreducible representations of an LM-decomposable Lie algebra. Infinite-dimensional variant of Lie's Theorem on a complex Banach space.- x 23 The associative envelope of a Lie algebra of quasinilpotent operators.- x 24 Commutativity modulo the Jacobson radical.- Notes.- IV Spectral Theory for Solvable Lie Algebras of Operators.- x 25 Spectral theory for representations of Lie algebras.- x 26 Spectral theory for systems of operators generating nilpotent Lie algebras.- x 27 The Cartan-Taylor spectrum of a locally solvable Lie algebra of operators.- x 28 Lie ideals of generalized spectral operators.- Notes.- V Semisimple Lie Algebras of Operators.- x 29 Lie subalgebra with involution consisting of bounded operators on a complex Banach space. Normal elements given by a space of self-adjoint operators.- x 30 Individual spectral properties in ideally finite semisimple Lie algebras of operators.- x 31 Semisimple Lie algebras of compact quasinilpotent operators.- Notes.- List of Symbols.