Hilbert Space Operators

by Kubrusly, Carlos S.

• ISBN13: 9780817632427
• ISBN10: 0817632425
• Format: Paperback
• Copyright: 2003-08-01
• Publisher: Birkhauser

Summary

This self-contained treatment of bounded linear operators on a Hilbert space provides an examination of the theory from a problem-solving viewpoint. Each chapter interweaves theoretical results with a number of problems, ranging from simple yet instructive exercises to open questions at the forefront of current research; complete solutions to all stated problems are provided. Written in a motivating and rigorous style, the text covers much of the classical theory: it begins with the basics of invariant subspaces, linear operators, convergence, shifts, and decompositions, and then proceeds to hyponormal operators, spectral properties, and paranormal and quasireducible operators. The book concludes with a detailed presentation of the Lomonosov Theorem on nontrivial hyperinvariant subspaces for compact operators. Some knowledge of elementary functional analysis and a familiarity with the basics of operator theory are all that is required. While this problem-solving approach to the study of Hilbert space operators is primarily aimed at graduate students, it will benefit researchers and working scientists as well, given the far-reaching applications of the subject to pure and applied mathematics, physics, engineering, economics, and statistics.

Table of Contents

Preface	vii
1 Invariant Subspaces	1	(12)
Problem 1.1 Closure	2	(1)
Problem 1.2 Kernel and Range	2	(1)
Problem 1.3 Null Product	3	(1)
Problem 1.4 Operator Equation	3	(1)
Problem 1.5 Nilpotent and Algebraic	3	(1)
Problem 1.6 Polynomials	4	(1)
Problem 1.7 Totally Cyclic	5	(1)
Problem 1.8 Densely Intertwined	5	(1)
Problem 1.9 Hyperinvariant	6	(1)
Problem 1.10 Quasiaffine Transform	6	(1)
Solutions	7	(6)
2 Hilbert Space Operators	13	(10)
Problem 2.1 Adjoint	14	(1)
Problem 2.2 Nonnegativs	14	(1)
Problem 2.3 Contraction	15	(1)
Problem 2.4 Normal	15	(1)
Problem 2.5 Isometry	15	(1)
Problem 2.6 Unitary	16	(2)
Problem 2.7 Projection	18	(1)
Problem 2.8 Mutually Orthogonal	18	(1)
Problem 2.9 Increasing	18	(1)
Solutions	19	(4)
3 Convergence and Stability	23	(10)
Problem 3.1 Diagonal	24	(1)
Problem 3.2 Product	25	(1)
Problem 3.3 * -Preserving	25	(1)
Problem 3.4 Nonnegativs	26	(1)
Problem 3.5 Monotone	26	(1)
Problem 3.6 Self-Adjoint	26	(1)
Problem 3.7 Commutant	27	(1)
Problem 3.8 Convex Cone	27	(1)
Problem 3.9 Absolute Value	28	(1)
Solutions	28	(5)
4 Reducing Subspaces	33	(8)
Problem 4.1 T-Invariant	34	(1)
Problem 4.2 Matrix Form	34	(1)
Problem 4.3 T*-Invariant	35	(1)
Problem 4.4 T and T*-Invariant	35	(1)
Problem 4.5 Commuting with T and T*	35	(1)
Problem 4.6 Reducible	35	(1)
Problem 4.7 Restriction	36	(1)
Problem 4.8 Direct Sum	36	(1)
Problem 4.9 Unitarily Equivalent	36	(1)
Problem 4.10 Unitary Restriction	36	(2)
Solutions	38	(3)
5 Shifts	41	(10)
Problem 5.1 Unilateral	42	(2)
Problem 5.2 Bilateral	44	(1)
Problem 5.3 Multiplicity	44	(1)
Problem 5.4 Unitarily Equivalent	44	(1)
Problem 5.5 Reducible	44	(1)
Problem 5.6 Irreducible	44	(1)
Problem 5.7 Rotation	44	(1)
Problem 5.8 Riemann-Lebesgue Lemma	45	(1)
Problem 5.9 Weighted Shift	45	(1)
Problem 5.10 Nonnegativs Weights	45	(1)
Solutions	46	(5)
6 Decompositions	51	(14)
Problem 6.1 Strong Limit	51	(1)
Problem 6.2 Projection	52	(1)
Problem 6.3 Kernels	52	(1)
Problem 6.4 Kernel Decomposition	52	(1)
Problem 6.5 Intertwined to Isometry	52	(1)
Problem 6.6 Dual Limits	53	(1)
Problem 6.7 Nagy-Foia -Langer Decomposition	53	(1)
Problem 6.8 von Neumann-Wold Decomposition	54	(1)
Problem 6.9 Another Decomposition	54	(1)
Problem 6.10 Foguel Decomposition	55	(1)
Problem 6.11 Isometry	55	(1)
Problem 6.12 Coisometry	55	(1)
Problem 6.13 Strongly Stable	55	(1)
Problem 6.14 Property PF	55	(1)
Problem 6.15 Direct Summand	56	(1)
Solutions	56	(9)
7 Hyponormal Operators	65	(10)
Problem 7.1 Quasinormal	65	(1)
Problem 7.2 Strong Stability	65	(1)
Problem 7.3 Hyponormal	66	(1)
Problem 7.4 Direct Proof	66	(1)
Problem 7.5 Invariant Subspace	66	(1)
Problem 7.6 Restriction	67	(1)
Problem 7.7 Normal	67	(1)
Problem 7.8 Roots of Powers	67	(1)
Problem 7.9 Normalord	68	(1)
Problem 7.10 Power Inequality	68	(1)
Problem 7.11 Unitarily Equivalent	68	(1)
Problem 7.12 Subnormal	69	(1)
Problem 7.13 Not Subnormal	69	(1)
Problem 7.14 Distinct Weights	69	(1)
Solutions	69	(6)
8 Spectral Properties	75	(18)
Problem 8.1 Spectrum	77	(1)
Problem 8.2 Eigenspace	77	(1)
Problem 8.3 Examples	78	(1)
Problem 8.4 Residual Spectrum	79	(1)
Problem 8.5 Weighted Shift	79	(1)
Problem 8.6 Uniform Stability	79	(1)
Problem 8.7 Finite Rank	80	(1)
Problem 8.8 Stability for Compact	80	(1)
Problem 8.9 Continuous Spectrum	81	(1)
Problem 8.10 Compact Contraction	81	(1)
Problem 8.11 Normal	82	(1)
Problem 8.12 Square Root	83	(1)
Problem 8.13 Fuglede Theorem	84	(1)
Problem 8.14 Quasinormal	84	(1)
Problem 8.15 Fuglede-Putnam Theorem	84	(1)
Problem 8.16 Reducible	84	(1)
Solutions	85	(8)
9 Paranormal Operators	93	(16)
Problem 9.1 Quasihyponormal	93	(1)
Problem 9.2 Semi-quasihyponormal	93	(1)
Problem 9.3 Paranormal	94	(1)
Problem 9.4 Square of Paranormal	94	(1)
Problem 9.5 Alternative Definition	94	(1)
Problem 9.6 Unitarily Equivalent	94	(1)
Problem 9.7 Weighted Shift	94	(1)
Problem 9.8 Equivalences	95	(1)
Problem 9.9 Not Paranormal	96	(1)
Problem 9.10 Projection ® Nilpotent	96	(1)
Problem 9.11 Shifted Operators	97	(1)
Problem 9.12 Shifted Projections	97	(1)
Problem 9.13 Shifted Self-Adjoints	97	(1)
Problem 9.14 Examples	98	(1)
Problem 9.15 Hyponormal	98	(1)
Problem 9.16 Invertible	99	(1)
Problem 9.17 Paranormal Inequality	99	(1)
Problem 9.18 Normalord	99	(1)
Problem 9.19 Cohyponormal	99	(1)
Problem 9.20 Strongly Stable	99	(1)
Problem 9.21 Quasinormal	99	(1)
Solutions	99	(10)
10 Proper Contractions	109	(8)
Problem 10.1 Equivalences	109	(1)
Problem 10.2 Diagonal	109	(1)
Problem 10.3 Compact	110	(1)
Problem 10.4 Adjoint	110	(1)
Problem 10.5 Paranormal	110	(1)
Problem 10.6 Nagy-Foia Classes	110	(1)
Problem 10.7 Weakly Stable	111	(1)
Problem 10.8 Hyponormal	111	(1)
Problem 10.9 Subnormal	111	(1)
Problem 10.10 Quasinormal	111	(1)
Problem 10.11 Direct Proof	111	(1)
Problem 10.12 Invariant Subspace	112	(1)
Solutions	112	(5)
11 Quasireducible Operators	117	(12)
Problem 11.1 Alternative Definition	118	(1)
Problem 11.2 Basic Properties	118	(1)
Problem 11.3 Nilpotent	118	(1)
Problem 11.4 Index 2	118	(1)
Problem 11.5 Higher Indices	118	(1)
Problem 11.6 Product	118	(1)
Problem 11.7 Unitarily Equivalent	119	(1)
Problem 11.8 Similarity	119	(1)
Problem 11.9 Unilateral Shift	119	(1)
Problem 11.10 Isometry	119	(1)
Problem 11.11 Quasinormal	119	(1)
Problem 11.12 Weighted Shift	119	(1)
Problem 11.13 Subnormal	119	(1)
Problem 11.14 Commutator	120	(1)
Problem 11.15 Reducible	120	(1)
Problem 11.16 Normal	120	(1)
Solutions	121	(8)
12 The Lomonosov Theorem	129	(14)
Problem 12.1 Hilden's Proof	129	(3)
Problem 12.2 Lomonosov Lemma	132	(1)
Problem 12.3 Lomonosov Theorem	133	(1)
Problem 12.4 Extension	134	(1)
Problem 12.5 Quasireducible	135	(1)
Problem 12.6 Hyponormal	135	(1)
Solutions	136	(7)
References	143	(4)
Index	147