Hilbert Space Operators, 2003
A Problem Solving Approach

This is a problem book on Hilbert space operators (Le. , on bounded linear transformations of a Hilbert space into itself) where theory and problems are investigated together. We tre!l:t only a part of the so-called single operator theory. Selected prob­ lems, ranging from standard textbook material to points on the boundary of the subject, are organized into twelve chapters. The book begins with elementary aspects of Invariant Subspaces for operators on Banach spaces 1. Basic properties of Hilbert Space Operators are introduced in in Chapter Chapter 2, Convergence and Stability are considered in Chapter 3, and Re­ ducing Subspaces is the theme of Chapter 4. Primary results about Shifts on Hilbert space comprise Chapter 5. These are introductory chapters where the majority of the problems consist of auxiliary results that prepare the ground for the next chapters. Chapter 6 deals with Decompositions for Hilbert space contractions, Chapter 7 focuses on Hyponormal Operators, and Chapter 8 is concerned with Spectral Properties of operators on Banach and Hilbert spaces. The next three chapters (as well as Chapter 6) carry their subjects from an introductory level to a more advanced one, including some recent results. Chapter 9 is about Paranormal Operators, Chapter 10 covers Proper Contractions, and Chapter 11 searches through Quasi­ reducible Operators. The final Chapter 12 commemorates three decades of The Lomonosov Theorem on nontrivial hyperinvariant subspaces for compact operators.

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.

...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 reference book in a broad sense...."-Mathematical Reviews