Operator Commutation Relations

In his Retiring Presidential address, delivered before the Annual Meeting of The American Mathematical Society on December, 1948, the late Professor Einar Hille spoke on his recent results on the Lie theory of semigroups of linear transformations, . . . "So far only commutative operators have been considered and the product law . . . is the simplest possible. The non-commutative case has resisted numerous attacks in the past and it is only a few months ago that any headway was made with this problem. I shall have the pleasure of outlining the new theory here; it is a blend of the classical theory of Lie groups with the recent theory of one-parameter semigroups. " The list of references in the subsequent publication of Hille's address (Bull. Amer. Math .. Soc. 56 (1950)) includes pioneering papers of I. E. Segal, I. M. Gelfand, and K. Yosida. In the following three decades the subject grew tremendously in vitality, incorporating a number of different fields of mathematical analysis. Early papers of V. Bargmann, I. E. Segal, L. G~ding, Harish-Chandra, I. M. Singer, R. Langlands, B. Konstant, and E. Nelson developed the theoretical basis for later work in a variety of different applications: Mathematical physics, astronomy, partial differential equations, operator algebras, dynamical systems, geometry, and, most recently, stochastic filtering theory. As it turned out, of course, the Lie groups, rather than the semigroups, provided the focus of attention.

Contenu

I: Some Main Results on Commutator Identities.- 1. Introduction and Survey.- 1A General Objectives of the Monograph.- 1B Contact with Prior Literature.- 1C The Main Results in Commutation Theory.- 1D The Main Results in Exponentiation Theory.- 1E Results on (Semi) Group-invariant C?-domains.- 1F Typical Applications of Commutation Theory.- 1G Typical Applications of Exponentiation Theory.- 2. The Finite-Dimensional Commutation Condition.- 2A Implications of Finite-Dimensionality in Commutation Theory.- 2B Examples involving Differential Operators.- 2C Examples from Universal and Operator Enveloping Algebras.- 2D Relaxing the Finite-Dimensionality Condition.- II: Commutation Relations and Regularity Properties for Operators in the Enveloping Algebra of Representations of Lie Groups.- 3. Domain Regularity and Semigroup Commutation Relations.- 3A Lie Algebras of Continuous Operators.- 3B Semigroups and Ad-Orbits.- 3C Variations upon the Regularity Condition.- 3D Infinite-Dimensional OA(B).- 4. Invariant-Domain Commutation Theory applied to the Mass-Splitting Principle.- 4A Global Invariance/Regularity for Heat-Type Semigroups.- 4B Formulation of the Generalized Mass-Splitting Theorem.- 4C The Mass-Operator as a Commuting Difference of Sub-Laplacians.- 4D Remarks on General Minkowskian Observables.- 4E Fourier Transform Calculus and Centrality of Isolated Projections.- III: Conditions for a System of Unbounded Operators to Satisfy a given Commutation Relation.- 5. Graph-Density applied to Resolvent Commutation, and Operational Calculus.- 5A Augmented Spectra and Resolvent Commutation Relations.- 5B Commutation Relations on D1.- 5C Analytic Continuation of Commutation Relations.- 5D Commutation Relations for the Holomorphic Operational Calculus.- 6. Graph-Density Applied to Semigroup Commutation Relations.- 6A Semigroup Commutation Relations with a Closable Basis.- 6B Variants of Sections 5B and 6A for General M.- 6C Automatic Availability of a Closable Basis.- 6D Remarks on Operational Calculi.- 7. Construction of Globally Semigroup-invariant C?-domains.- 7A Fréchet C?-domains in Banach Spaces.- 7B The Extrinsic Two-Operator Case.- 7C The Lie Algebra Case.- 7D C?-action of Resolvents, Projections, and Operational Calculus.- IV: Conditions for a Lie Algebra of Unbounded Operators to Generate a Strongly Continuous Representation of the Lie Group.- 8. Integration of Smooth Operator Lie Algebras.- 8A Smooth Lie Algebras and Differentiable Representations.- 8B Applications in C?-vector spaces.- 9. Exponentiation and Bounded Perturbation of Operator Lie Algebras.- 9A Discussion of Exponentiation Theorems and Applications.- 9B Proofs of the Theorems.- 9C Phillips Perturbations of Operator Lie Algebras and Analytic Continuation of Group Representations.- 9D Semidirect Product Perturbations.- Appendix to Part IV.- V: Lie Algebras of Vector Fields on Manifolds.- 10. Applications of Commutation Theory to Vector-Field Lie Algebras and Sub- Laplacians on Manifolds.- 10A Exponentials versus Geometric Integrals of Vector-Field Lie Algebras.- 10B Exponentiation on Lp spaces.- 10C Sub-Laplacians on Manifolds.- 10D Solution Kernels on Manifolds.- VI: Derivations on Modules of Unbounded Operators with Applications to Partial Differential Operators on Riemann Surfaces.- 11. Rigorous Analysis of Some Commutator Identities for Physical Observables.- 11A Variations upon the Graph-Density and Kato Conditions.- 11B Various forms of Strong Commutativity.- 11C Nilpotent Commutation Relations of Generalized Heisenberg-Weyl Type.- Appendix to Part VI.- VII: Lie Algebras of Unbounded Operators: Perturbation Theory, and Analytic Continuation of s?(2, ?)-Modules.- 12. Exponentiation and Analytic Continuation of Heisenberg-Matrix Representations for s?(2, ?).- 12A Connections to the Theory of TCI Representations of Semisimple Groups on Banach Spaces.- 12B The Graph-Density Condition and Base-Point Exponentials.- 12C C?-integrals and Smeared Exponentials on ?p.- 12D The Operators A0, A1 and A2.- 12E Compact and Phillips Perturbations.- 12F Perturbations and Analytic Continuation of Smeared Representations.- 12G Irreducibility, Equivalences, Unitarity, and Single-Valuedness.- 12H Perturbation and Reduction Properties of Other Analytic Series.- 12I A Counter-Theorem on Group-Invariant Domains.- Appendix to Part VII.- General Appendices.- Appendix A. The Product Rule for Differentiable Operator Valued Mappings.- Appendix B. A Review of Semigroup Folklore, and Integration in Locally Convex Spaces.- Appendix C. The Square of an Infinitesimal Group Generator.- Appendix E. Compact Perturbations of Semigroups.- Appendix G. Bounded Elements in Operator Lie Algebras.- References.- References to Ouotations.- List of Symbols.