Operator Commutation Relations

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.