Invariant Subspaces

by Heydar Radjavi, Peter Rosenthal (Editor)

0. Introduction and Preliminaries.- 0.1 Hilbert Space.- 0.2 Invariant Subspaces.- 0.3 Spectra of Operators.- 0.4 Linear Operator Equations.- 0.5 Additional Propositions.- 0.6 Notes and Remarks.- 1. Normal Operators.- 1.1 Preliminaries.- 1.2 Compact Normal Operators.- 1.3 Spectral Theorem-First Form.- 1.4 Spectral Theorem-Second Form.- 1.5 Fuglede's Theorem.- 1.6 The Algebra ??.- 1.7 The Functional Calculus.- 1.8 Completely Normal Operators.- 1.9 Additional Propositions.- 1.10 Notes and Remarks.- 2. Analytic Functions of Operators.- 2.1 The Functional Calculus.- 2.2 The Riesz Decomposition Theorem.- 2.3 Invariant Subspaces of Analytic Functions of Operators.- 2.4 Additional Propositions.- 2.5 Notes and Remarks.- 3. Shift Operators.- 3.1 Shifts of Multiplicity 1.- 3.2 Invariant Subspaces of Shifts of Multiplicity 1.- 3.3 Shifts of Arbitrary Multiplicity.- 3.4 Invariant Subspaces of Shifts.- 3.5 Parts of Shifts.- 3.6 Additional Propositions.- 3.7 Notes and Remarks.- 4. Examples of Invariant Subspace Lattices.- 4.1 Preliminaries.- 4.2 Algebraic Operators.- 4.3 Lattices of Normal Operators.- 4.4 Two Unicellular Operators.- 4.5 Direct Products of Attainable Lattices.- 4.6 Attainable Ordinal Sums.- 4.7 Transitive Lattices.- 4.8 Additional Propositions.- 4.9 Notes and Remarks.- 5. Compact Operators.- 5.1 Existence of Invariant Subspaces.- 5.2 Normality and Lat A.- 5.3 Spectrum and Lat A.- 5.4 Lattices of Compact Operators.- 5.5 Additional Propositions.- 5.6 Notes and Remarks.- 6. Existence of Invariant and Hyperinvariant Subspaces.- 6.1 Operators on Other Spaces.- 6.2 Perturbations of Normal Operators.- 6.3 Quasi-similarity and Invariant Subspaces.- 6.4 Hyperinvariant Subspaces.- 6.5 Additional Propositions.- 6.6 Notes and Remarks.- 7. Certain Results on von Neumann Algebras.- 7.1 Preliminaries.- 7.2 Commutants.- 7.3 The Algebra ? (?).- 7.4 Abelian von Neumann Algebras.- 7.5 The Class of n-normal Operators.- 7.6 Additional Propositions.- 7.7 Notes and Remarks.- 8. Transitive Operator Algebras.- 8.1 Strictly Transitive Algebras.- 8.2 Partial Solutions of the Transitive Algebra Problem.- 8.3 Generators of ? (?).- 8.4 Additional Propositions.- 8.5 Notes and Remarks.- 9. Algebras Associated with Invariant Subspaces.- 9.1 Reductive Algebras.- 9.2 Reflexive Operator Algebras.- 9.3 Triangular Operator Algebras.- 9.4 Additional Propositions.- 9.5 Notes and Remarks.- 10. Some Unsolved Problems.- 10.1 Normal Operators.- 10.2 Attainable Lattices.- 10.3 Existence of Invariant Subspaces.- 10.4 Reducing Subspaces and von Neumann Algebras.- 10.5 Transitive and Reductive Algebras.- 10.6 Reflexive Algebras.- 10.7 Triangular Algebras.- References.- List of Symbols.- Author Index.

240 pages, Paperback

First published November 19, 1973