9780486428222

Invariant Subspaces

by ;
  • ISBN13:

    9780486428222

  • ISBN10:

    0486428222

  • Edition: 2nd
  • Format: Paperback
  • Copyright: 2003-07-15
  • Publisher: Dover Publications

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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

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