Ebook: Operator Commutation Relations: Commutation Relations for Operators, Semigroups, and Resolvents with Applications to Mathematical Physics and Representations of Lie Groups

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.

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Content:
Front Matter....Pages i-xviii
Front Matter....Pages 1-2
Introduction and Survey....Pages 3-36
The Finite-Dimensional Commutation Condition....Pages 37-53
Front Matter....Pages 55-59
Domain Regularity and Semigroup Commutation Relations....Pages 60-76
Invariant-Domain Commutation Theory Applied to the Mass-Splitting Principle....Pages 77-97
Front Matter....Pages 99-107
Graph-Density Applied to Resolvent Commutation, and Operational Calculus....Pages 108-130
Graph-Density Applied to Semigroup Commutation Relations....Pages 131-149
Construction of Globally Semigroup-Invariant C? -Domains....Pages 150-169
Front Matter....Pages 171-176
Integration of Smooth Operator Lie Algebras....Pages 177-193
Exponentiation and Bounded Perturbation of Operator Lie Algebras....Pages 194-226
Back Matter....Pages 227-232
Front Matter....Pages 233-239
Applications of Commutation Theory to Vector-Field Lie Algebras and Sub-Laplacians on Manifolds....Pages 240-274
Front Matter....Pages 275-278
Rigorous Analysis of Some Commutator Identities for Physical Observables....Pages 279-319
Back Matter....Pages 320-327
Front Matter....Pages 329-334
Exponentiation and Analytic Continuation of Heisenberg-Matrix Representations for s?(2,?)....Pages 335-431
Back Matter....Pages 432-436
Back Matter....Pages 437-493

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Content:
Front Matter....Pages i-xviii
Front Matter....Pages 1-2
Introduction and Survey....Pages 3-36
The Finite-Dimensional Commutation Condition....Pages 37-53
Front Matter....Pages 55-59
Domain Regularity and Semigroup Commutation Relations....Pages 60-76
Invariant-Domain Commutation Theory Applied to the Mass-Splitting Principle....Pages 77-97
Front Matter....Pages 99-107
Graph-Density Applied to Resolvent Commutation, and Operational Calculus....Pages 108-130
Graph-Density Applied to Semigroup Commutation Relations....Pages 131-149
Construction of Globally Semigroup-Invariant C? -Domains....Pages 150-169
Front Matter....Pages 171-176
Integration of Smooth Operator Lie Algebras....Pages 177-193
Exponentiation and Bounded Perturbation of Operator Lie Algebras....Pages 194-226
Back Matter....Pages 227-232
Front Matter....Pages 233-239
Applications of Commutation Theory to Vector-Field Lie Algebras and Sub-Laplacians on Manifolds....Pages 240-274
Front Matter....Pages 275-278
Rigorous Analysis of Some Commutator Identities for Physical Observables....Pages 279-319
Back Matter....Pages 320-327
Front Matter....Pages 329-334
Exponentiation and Analytic Continuation of Heisenberg-Matrix Representations for s?(2,?)....Pages 335-431
Back Matter....Pages 432-436
Back Matter....Pages 437-493
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