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.

Inhalt
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* ? TT 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 Spectralrepresentation 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.