A Problem Solving Approach
Buch, Englisch, 149 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 265 g
ISBN: 978-0-8176-3242-7
Verlag: Birkhäuser Boston
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.
Zielgruppe
Research
Autoren/Hrsg.
Fachgebiete
Weitere Infos & Material
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.
