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Full Description
In recent years there has been a large amount of work on invariant subspaces, motivated by interest in the structure of non-self-adjoint of the results have been obtained in operators on Hilbert space. Some the context of certain general studies: the theory of the characteristic operator function, initiated by Livsic; the study of triangular models by Brodskii and co-workers; and the unitary dilation theory of Sz. Nagy and Foia!? Other theorems have proofs and interest independent of any particular structure theory. Since the leading workers in each of the structure theories have written excellent expositions of their work, (cf. Sz.-Nagy-Foia!? [1], Brodskii [1], and Gohberg-Krein [1], [2]), in this book we have concentrated on results independent of these theories. We hope that we have given a reasonably complete survey of such results and suggest that readers consult the above references for additional information. The table of contents indicates the material covered. We have restricted ourselves to operators on separable Hilbert space, in spite of the fact that most of the theorems are valid in all Hilbert spaces and many hold in Banach spaces as well. We felt that this restriction was sensible since it eases the exposition and since the separable-Hilbert space case of each of the theorems is generally the most interesting and potentially the most useful case.
Contents
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 NeumannAlgebras.- 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.
