Invariant Subspaces

by Radjavi, Heydar; Rosenthal, Peter

• ISBN13: 9780486428222
• ISBN10: 0486428222
• Edition: 2nd
• Format: Paperback
• Copyright: 2003-07-15
• Publisher: Dover Publications

Summary

This broad survey spans a wealth of studies on invariant subspaces, focusing on operators on separable Hilbert space. Largely self-contained, it requires only a working knowledge of measure theory, complex analysis, and elementary functional analysis. Subjects include normal operators, analytic functions of operators, shift operators, examples of invariant subspace lattices, compact operators, and the existence of invariant and hyperinvariant subspaces. Additional chapters cover certain results on von Neumann algebras, transitive operator algebras, algebras associated with invariant subspaces, and a selection of unsolved problems. 1973 ed. New appendix on recent developments.

Table of Contents

Introduction and Preliminaries	1	(11)
Hilbert Space	1	(1)
Invariant Subspaces	2	(2)
Spectra of Operators	4	(3)
Linear Operator Equations	7	(2)
Additional Propositions	9	(1)
Notes and Remarks	10	(1)
Normal Operators	11	(15)
Preliminaries	11	(1)
Compact Normal Operators	12	(1)
Spectral Theorem---First Form	13	(3)
Spectral Theorem---Second Form	16	(3)
Fuglede's Theorem	19	(2)
The Algebra L∞	21	(1)
The Functional Calculus	21	(1)
Completely Normal Operators	22	(1)
Additional Propositions	23	(2)
Notes and Remarks	25	(1)
Analytic Functions of Operators	26	(10)
The Functional Calculus	26	(5)
The Riesz Decomposition Theorem	31	(1)
Invariant Subspaces of Analytic Functions of Operators	32	(1)
Additional Propositions	33	(1)
Notes and Remarks	34	(2)
Shift Operators	36	(24)
Shifts of Multiplicity 1	36	(2)
Invariant Subspaces of Shifts of Multiplicity 1	38	(8)
Shifts of Arbitrary Multiplicity	46	(4)
Invariant Subspaces of Shifts	50	(3)
Parts of Shifts	53	(4)
Additional Propositions	57	(2)
Notes and Remarks	59	(1)
Examples of Invariant Subspace Lattices	60	(24)
Preliminaries	60	(2)
Algebraic Operators	62	(2)
Lattices of Normal Operators	64	(2)
Two Unicellular Operators	66	(6)
Direct Products of Attainable Lattices	72	(3)
Attainable Ordinal Sums	75	(3)
Transitive Lattices	78	(3)
Additional Propositions	81	(1)
Notes and Remarks	82	(2)
Compact Operators	84	(11)
Existence of Invariant Subspaces	84	(3)
Normality and Lat A	87	(1)
Spectrum and Lat A	88	(2)
Lattices of Compact Operators	90	(2)
Additional Propositions	92	(1)
Notes and Remarks	93	(2)
Existence of Invariant and Hyperinvariant Subspaces	95	(22)
Operators on Other Spaces	95	(2)
Perturbations of Normal Operators	97	(11)
Quasi-similarity and Invariant Subspaces	108	(2)
Hyperinvariant Subspaces	110	(3)
Additional Propositions	113	(1)
Notes and Remarks	114	(3)
Certain Results on von Neumann Algebras	117	(21)
Preliminaries	117	(1)
Commutants	118	(2)
The Algebra B (H)	120	(2)
Abelian von Neumann Algebras	122	(5)
The Class of n-normal Operators	127	(9)
Additional Propositions	136	(1)
Notes and Remarks	136	(2)
Transitive Operator Algebras	138	(29)
Strictly Transitive Algebras	138	(4)
Partial Solutions of the Transitive Algebra Problem	142	(18)
Generators of B(H)	160	(3)
Additional Propositions	163	(1)
Notes and Remarks	164	(3)
Algebras Associated with Invariant Subspaces	167	(25)
Reductive Algebras	167	(10)
Reflexive Operator Algebras	177	(8)
Triangular Operator Algebras	185	(3)
Additional Propositions	188	(1)
Notes and Remarks	189	(3)
Some Unsolved Problems	192	(7)
Normal Operators	192	(1)
Attainable Lattices	193	(1)
Existence of Invariant Subspaces	194	(1)
Reducing Subspaces and von Neumann Algebras	195	(1)
Transitive and Reductive Algebras	196	(1)
Reflexive Algebras	197	(1)
Triangular Algebras	198	(1)
References	199	(12)
List of Symbols	211	(1)
Author Index	212	(3)
Subject Index	215	(16)
Some Subsequent Developments	221	(10)
Normal Operators	221	(1)
Attainable Lattices	222	(1)
Existence of Invariant Subspaces	222	(3)
Reducing Subspaces and von Neumann algebras	225	(1)
Transitive and Reductive Algebras	225	(2)
Reflexive Algebras	227	(1)
Triangular Operator Algebras	228	(3)
Additional References	231