Invariant Subspaces

by Heydar Radjavi, Peter Rosenthal

Overview

Broad survey focuses on operators on separable Hilbert spaces. Topics include normal operators, analytic functions of operators, shift operators, invariant subspace lattices, compact operators, invariant and hyperinvariant subspaces, von Neumann algebras, transitive operator algebras, and algebras associated with invariant subspaces. 1973 edition. New Appendix on Recent Developments.

Product Details

ISBN-13:	9780486153025
Publisher:	Dover Publications
Publication date:	11/30/2011
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	256
File size:	21 MB
Note:	This product may take a few minutes to download.

Table of Contents

Chapter 0.	Introduction and Preliminaries	1
0.1	Hilbert Space	1
0.2	Invariant Subspaces	2
0.3	Spectra of Operators	4
0.4	Linear Operator Equations	7
0.5	Additional Propositions	9
0.6	Notes and Remarks	10
Chapter 1.	Normal Operators	11
1.1	Preliminaries	11
1.2	Compact Normal Operators	12
1.3	Spectral Theorem--First Form	13
1.4	Spectral Theorem--Second Form	16
1.5	Fuglede's Theorem	19
1.6	The Algebra L[superscript infinity]	21
1.7	The Functional Calculus	21
1.8	Completely Normal Operators	22
1.9	Additional Propositions	23
1.10	Notes and Remarks	25
Chapter 2.	Analytic Functions of Operators	26
2.1	The Functional Calculus	26
2.2	The Riesz Decomposition Theorem	31
2.3	Invariant Subspaces of Analytic Functions of Operators	32
2.4	Additional Propositions	33
2.5	Notes and Remarks	34
Chapter 3.	Shift Operators	36
3.1	Shifts of Multiplicity 1	36
3.2	Invariant Subspaces of Shifts of Multiplicity 1	38
3.3	Shifts of Arbitrary Multiplicity	46
3.4	Invariant Subspaces of Shifts	50
3.5	Parts of Shifts	53
3.6	Additional Propositions	57
3.7	Notes and Remarks	59
Chapter 4.	Examples of Invariant Subspace Lattices	60
4.1	Preliminaries	60
4.2	Algebraic Operators	62
4.3	Lattices of Normal Operators	64
4.4	Two Unicellular Operators	66
4.5	Direct Products of Attainable Lattices	72
4.6	Attainable Ordinal Sums	75
4.7	Transitive Lattices	78
4.8	Additional Propositions	81
4.9	Notes and Remarks	82
Chapter 5.	Compact Operators	84
5.1	Existence of Invariant Subspaces	84
5.2	Normality and Lat A	87
5.3	Spectrum and Lat A	88
5.4	Lattices of Compact Operators	90
5.5	Additional Propositions	92
5.6	Notes and Remarks	93
Chapter 6.	Existence of Invariant and Hyperinvariant Subspaces	95
6.1	Operators on Other Spaces	95
6.2	Perturbations of Normal Operators	97
6.3	Quasi-similarity and Invariant Subspaces	108
6.4	Hyperinvariant Subspaces	110
6.5	Additional Propositions	113
6.6	Notes and Remarks	114
Chapter 7.	Certain Results on von Neumann Algebras	117
7.1	Preliminaries	117
7.2	Commutants	118
7.3	The Algebra B (H)	120
7.4	Abelian von Neumann Algebras	122
7.5	The Class of n-normal Operators	127
7.6	Additional Propositions	136
7.7	Notes and Remarks	136
Chapter 8.	Transitive Operator Algebras	138
8.1	Strictly Transitive Algebras	138
8.2	Partial Solutions of the Transitive Algebra Problem	142
8.3	Generators of B (H)	160
8.4	Additional Propositions	163
8.5	Notes and Remarks	164
Chapter 9.	Algebras Associated with Invariant Subspaces	167
9.1	Reductive Algebras	167
9.2	Reflexive Operator Algebras	177
9.3	Triangular Operator Algebras	185
9.4	Additional Propositions	188
9.5	Notes and Remarks	189
Chapter 10.	Some Unsolved Problems	192
10.1	Normal Operators	192
10.2	Attainable Lattices	193
10.3	Existence of Invariant Subspaces	194
10.4	Reducing Subspaces and von Neumann Algebras	195
10.5	Transitive and Reductive Algebras	196
10.6	Reflexive Algebras	197
10.7	Triangular Algebras	198
	References	199
	List of Symbols	211
	Author Index	212
	Subject Index	215
Chapter 11.	Some Subsequent Developments	221
11.1	Normal Operators	221
11.2	Attainable Lattices	222
11.3	Existence of Invariant Subspaces	222
11.4	Reducing Subspaces and von Neumann algebras	225
11.5	Transitive and Reductive Algebras	225
11.6	Reflexive Algebras	227
11.7	Triangular Operator Algebras	228
	Additional References	231