Commutation Properties of Hilbert Space Operators and Related Topics (Paperback)

What could be regarded as the beginning of a theory of commutators AB - BA of operators A and B on a Hilbert space, considered as a dis­ cipline in itself, goes back at least to the two papers of Weyl [3] {1928} and von Neumann [2] {1931} on quantum mechanics and the commuta­ tion relations occurring there. Here A and B were unbounded self-adjoint operators satisfying the relation AB - BA = iI, in some appropriate sense, and the problem was that of establishing the essential uniqueness of the pair A and B. The study of commutators of bounded operators on a Hilbert space has a more recent origin, which can probably be pinpointed as the paper of Wintner [6] {1947}. An investigation of a few related topics in the subject is the main concern of this brief monograph. The ensuing work considers commuting or "almost" commuting quantities A and B, usually bounded or unbounded operators on a Hilbert space, but occasionally regarded as elements of some normed space. An attempt is made to stress the role of the commutator AB - BA, and to investigate its properties, as well as those of its components A and B when the latter are subject to various restrictions. Some applica­ tions of the results obtained are made to quantum mechanics, perturba­ tion theory, Laurent and Toeplitz operators, singular integral trans­ formations, and Jacobi matrices.

I. Commutators of bounded operators.- 1.1 Introduction.- 1.2 Structure of commutators of bounded operators.- 1.3 Commutators C = AB?BA with AC = CA.- 1.4 Multiplicative commutators.- 1.5 Commutators and numerical range.- 1.6 Some results on normal operators.- 1.7 Operator equation BX?XA= Y.- II. Commutators and spectral theory.- 2.1 Introduction.- 2.2 Spectral properties.- 2.3 Absolute continuity and measure of spectrum.- 2.4 Absolute continuity and numerical range.- 2.5 Higher order commutators.- 2.6 Further results on commutators and normal operators.- 2.7 Half-bounded operators and unitary equivalence.- 2.8 Half-boundedness and absolute continuity.- 2.9 Applications.- 2.10 Commutators of self-adjoint operators.- 2.11 Examples.- 2.12 More on non-negative perturbations and spectra.- 2.13 Commutators of self-adjoint operators.- 2.14 An application to quantum mechanics.- III. Semi-normal operators.- 3.1 Introduction.- 3.2 Structure properties.- 3.3 Spectrum of a semi-normal operator.- 3.4 Further spectral properties.- 3.5 An integral formula.- 3.6 Isolated parts of sp (T).- 3.7 Measure of sp (T).- 3.8 Zero measure of sp (T) and normality.- 3.9 Special products of self-adjoint operators.- 3.10 Resolvents of semi-normal operators.- 3.11 Semi-normal operators and arc spectra.- 3.12 TT* ? T*T of one-dimensional range.- 3.13 An example concerning T2.- 3.14 Subnormal operators.- IV. Commutation relations in quantum mechanics.- 4.1 Introduction.- 4.2 Unitary groups itP and eisQ.- 4.3 Von Neumann’s theorem.- 4.4 The equation AA* = A*A+I.- 4.5 The operators P and Q.- 4.6 Results of Rellich and Dixmier.- 4.7 Results of Tillmann.- 4.8 Results of Foia?, Gehér and Sz.-Nagy.- 4.9 A result of Kato.- 4.10 Results of Kristensen, Mejlbo and Poulsen.- 4.11 Systems with n(< ?) degrees of freedom.- 4.12 Anticommutation relations.- 4.13 General systems.- 4.14 A uniqueness theorem.- 4.15 Existence of the vacuum state.- 4.16 Self-adjointness of ?A?*A?.- 4.17 Remarks on commutators and the equations of motion.- V. Wave operators and unitary equivalence of self-adjoint operators.- 5.1 Introduction and a basic theorem.- 5.2 Schmidt and trace classes.- 5.3 Some lemmas.- 5.4 One-dimensional perturbations.- 5.5 Perturbations by operators of trace class.- 5.6 Invariance of wave operators.- 5.7 Generalizations.- 5.8 Applications to differential operators.- 5.9 A sufficient condition for the existence of W±(H1, H0).- 5.10 Hamiltonian operators.- 5.11 Existence of W± for the Hamiltonian case.- 5.12 A criterion for self-adjointness of perturbed operators.- 5.13 Existence and properties of wave and scattering operators.- 5.14 Stationary approach to scattering.- 5.15 Non-negative perturbations.- 5.16 Hamiltonians and non-negative perturbations.- 5.17 Remarks on unitary equivalence.- VI. Laurent and Toeplitz operators, singular integral operators and Jacobi matrices.- 6.1 Laurent and Toeplitz operators.- 6.2 A spectral inclusion theorem.- 6.3 A special Toeplitz matrix.- 6.4 Spectra of self-adjoint Toeplitz operators.- 6.5 Two lemmas.- 6.6 Analytic and coanalytic Toeplitz operators.- 6.7 Absolute continuity of Toeplitz operators.- 6.8 Spectral resolutions for certain Toeplitz operators.- 6.9 Some results for unbounded operators.- 6.10 Hilbert matrix.- 6.11 Singular integral operators.- 6.12 A(h, ?, E) with E bounded.- 6.13 The norm of A(0, ?, E).- 6.14 An estimate of meas sp (A(h, ?, E)).- 6.15 Remarks.- 6.16 Absolute continuity.- 6.17 Other singular integrals.- 6.18 Reducing spaces of A(0, ?, E).- 6.19 Estimates for ?? and ??.- 6.20 Spectral representation for A(0,1, (a, b)).- 6.21 Remarks on the spectra of singular integral operators.- 6.22 Jacobi matrices and absolute continuity.- Symbol Index.- Author Index.