Operator Commutation Relations

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.