Carlos S. Kubrusly

Hilbert Space Operators (eBook, PDF)

A Problem Solving Approach

• Format: PDF

Produktbeschreibung

This self-contained work on Hilbert space operators takes a problem-solving approach to the subject, combining theoretical results with a wide variety of exercises that range from the straightforward to the state-of-the-art. Complete solutions to all problems are provided. The text covers the basics of bounded linear operators on a Hilbert space and gradually progresses to more advanced topics in spectral theory and quasireducible operators.

Written in a motivating and rigorous style, the work has few prerequisites beyond elementary functional analysis, and will appeal to graduate students and researchers in mathematics, physics, engineering, and related disciplines.

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Produktdetails

• Verlag: Birkhäuser Boston
• Seitenzahl: 149
• Erscheinungstermin: 6. Dezember 2012
• Englisch
• ISBN-13: 9781461220640
• Artikelnr.: 44209603

Autorenporträt

Carlos Kubrusly was born in Rio de Janeiro in November 1947. He received his Ph.D. from the University of Warwick in 1976, held postdoctoral visiting research positions at the Universities of Warwick and Bonn, and has been a faculty member of Catholic University of Rio de Janeiro since 1972, where he became a full professor in 1988. Professor Kubrusly has published over a hundred scientific articles in international journals and proceedings of international conferences. He has participated in about forty scientific conferences, all with paper presentation, including plenary sections at the Toulouse IFAC Symposium in 1982 and at the Newport Beach SOTA Conference in 1998, and has also been a member of program committees of several international conferences.

Inhaltsangabe

1 Invariant Subspaces.- Problem 1.1 Closure.- Problem 1.2 Kernel and Range.- Problem 1.3 Null Product.- Problem 1.4 Operator Equation.- Problem 1.5 Nilpotent and Algebraic.- Problem 1.6 Polynomials.- Problem 1.7 Totally Cyclic.- Problem 1.8 Densely Intertwined.- Problem 1.9 Hyperinvariant.- Problem 1.10 Quasiaffine Transform.- Solutions.- 2 Hilbert Space Operators.- Problem 2.1 Adjoint.- Problem 2.2 Nonnegative.- Problem 2.3 Contraction.- Problem 2.4 Normal.- Problem 2.5 Isometry.- Problem 2.6 Unitary.- Problem 2.7 Projection.- Problem 2.8 Mutually Orthogonal.- Problem 2.9 Increasing.- Solutions.- 3 Convergence and Stability.- Problem 3.1 Diagonal.- Problem 3.2 Product.- Problem 3.3 * -Preserving.- Problem 3.4 Nonnegative.- Problem 3.5 Monotone.- Problem 3.6 Self-Adjoint.- Problem 3.7 Commutant.- Problem 3.8 Convex Cone.- Problem 3.9 Absolute Value.- Solutions.- 4 Reducing Subspaces.- Problem 4.1 T-Invariant.- Problem 4.2 Matrix Form.- Problem 4.3 T*-Invariant.- Problem 4.4 T and T*-Invariant.- Problem 4.5 Commuting with T and T*.- Problem 4.6 Reducible.- Problem 4.7 Restriction.- Problem 4.8 Direct Sum.- Problem 4.9 Unitarily Equivalent.- Problem 4.10 Unitary Restriction.- Solutions.- 5 Shifts.- Problem 5.1 Unilateral.- Problem 5.2 Bilateral.- Problem 5.3 Multiplicity.- Problem 5.4 Unitarily Equivalent.- Problem 5.5 Reducible.- Problem 5.6 Irreducible.- Problem 5.7 Rotation.- Problem 5.8 Riemann-Lebesgue Lemma.- Problem 5.9 Weighted Shift.- Problem 5.10 Nonnegative Weights.- Solutions.- 6 Decompositions.- Problem 6.1 Strong Limit.- Problem 6.2 Projection.- Problem 6.3 Kernels.- Problem 6.4 Kernel Decomposition.- Problem 6.5 Intertwined to Isometry.- Problem 6.6 Dual Limits.- Problem 6.7 Nagy-Foia?-Langer Decomposition.- Problem 6.8 von Neumann-Wold Decomposition.-Problem 6.9 Another Decomposition.- Problem 6.10 Foguel Decomposition.- Problem 6.11 Isometry.- Problem 6.12 Coisometry.- Problem 6.13 Strongly Stable.- Problem 6.14 Property PF.- Problem 6.15 Direct Summand.- Solutions.- 7 Hyponormal Operators.- Problem 7.1 Quasinormal.- Problem 7.2 Strong Stability.- Problem 7.3 Hyponormal.- Problem 7.4 Direct Proof.- Problem 7.5 Invariant Subspace.- Problem 7.6 Restriction.- Problem 7.7 Normal.- Problem 7.8 Roots of Powers.- Problem 7.9 Normaloid.- Problem 7.10 Power Inequality.- Problem 7.11 Unitarily Equivalent.- Problem 7.12 Subnormal.- Problem 7.13 Not Subnormal.- Problem 7.14 Distinct Weights.- Solutions.- 8 Spectral Properties.- Problem 8.1 Spectrum.- Problem 8.2 Eigenspace.- Problem 8.3 Examples.- Problem 8.4 Residual Spectrum.- Problem 8.5 Weighted Shift.- Problem 8.6 Uniform Stability.- Problem 8.7 Finite Rank.- Problem 8.8 Stability for Compact.- Problem 8.9 Continuous Spectrum.- Problem 8.10 Compact Contraction.- Problem 8.11 Normal.- Problem 8.12 Square Root.- Problem 8.13 Fuglede Theorem.- Problem 8.14 Quasinormal.- Problem 8.15 Fuglede-Putnam Theorem.- Problem 8.16 Reducible.- Solutions.- 9 Paranormal Operators.- Problem 9.1 Quasihyponormal.- Problem 9.2 Semi-quasihyponormal.- Problem 9.3 Paranormal.- Problem 9.4 Square of Paranormal.- Problem 9.5 Alternative Definition.- Problem 9.6 Unitarily Equivalent.- Problem 9.7 Weighted Shift.- Problem 9.8 Equivalences.- Problem 9.9 Not Paranormal.- Problem 9.10 Projection ? Nilpotent.- Problem 9.11 Shifted Operators.- Problem 9.12 Shifted Projections.- Problem 9.13 Shifted Seif-Adjoints.- Problem 9.14 Examples.- Problem 9.15 Hyponormal.- Problem 9.16 Invertible.- Problem 9.17 Paranormal Inequality.- Problem 9.18 Normaloid.- Problem 9.19 Cohyponormal.- Problem 9.20 StronglyStable.- Problem 9.21 Quasinormal.- Solutions.- 10 Proper Contractions.- Problem 10.1 Equivalences.- Problem 10.2 Diagonal.- Problem 10.3 Compact.- Problem 10.4 Adjoint.- Problem 10.5 Paranormal.- Problem 10.6 Nagy-Foia? Classes.- Problem 10.7 Weakly Stable.- Problem 10.8 Hyponormal.- Problem 10.9 Subnormal.- Problem 10.10 Quasinormal.- Problem 10.11 Direct Proof.- Problem 10.12 Invariant Subspace.- Solutions.- 11 Quasireducible Operators.- Problem 11.1 Alternative Definition.- Problem 11.2 Basic Properties.- Problem 11.3 Nilpotent.- Problem 11.4 Index 2.- Problem 11.5 Higher Indices.- Problem 11.6 Product.- Problem 11.7 Unitarily Equivalent.- Problem 11.8 Similarity.- Problem 11.9 Unilateral Shift.- Problem 11.10 Isometry.- Problem 11.11 Quasinormal.- Problem 11.12 Weighted Shift.- Problem 11.13 Subnormal.- Problem 11.14 Commutator.- Problem 11.15 Reducible.- Problem 11.16 Normal.- Solutions.- 12 The Lomonosov Theorem.- Problem 12.1 Hilden's Proof.- Problem 12.2 Lomonosov Lemma.- Problem 12.3 Lomonosov Theorem.- Problem 12.4 Extension.- Problem 12.5 Quasireducible.- Problem 12.6 Hyponormal.- Solutions.- References.

Rezensionen

"...the author treats many new subjects of operator theory for graduate students and mathematicians, i.e., quasihyponormal operators, paranormal operators, proper contractions, quasireducible operators, a detailed presentation of the Lomonosov theorem on nontrivial hyperinvariant subspaces for compact operators, etc. This book will be of benefit to graduate students (in mathematics, physics, engineering, economics, and statistics) and many mathematicians."

-Zentralblatt MATH

"...many totally new subjects are offered to the potential reader.... The author has great teaching experience reflected by the skillful selection of background material, the gradual statements of the problems, and the detailed presentation of the solutions. The book is intended [for] an audience mainly formed by various types of graduate students: in mathematics, statistics, physics and, perhaps, in engineering and economics. It can be useful even for working mathematicians, as a referencebook in a broad sense...."

-Mathematical Reviews