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Methods of noncommutative analysis: theory and applications. (English) Zbl 0876.47015
de Gruyter Studies in Mathematics. 22. Berlin: Walter de Gruyter. x, 373 p. (1995).
The monograph is devoted to the theory and applications of the advanced tools in modern mathematics: the calculus of noncommuting operators. This theory was started by R. Feynman for quantum electrodynamics. Real move in this theory was made later on by V. Maslov.
The monograph represents an attempt of introduction to noncommutative analysis and also gives to professionals in this area some new topics. The monograph consists of four chapters and two appendices.
Chapter I is the elementary introduction to noncommutative analysis. Some mathematical problems where different types of functions of noncommuting operators arise are described (T-exponentials, quantum mechanics, pertutation theory, Lie groups, and others). Main constructions and main definitions, basic properties of noncommuting operators are given. Problems and formulas of noncommutative differential calculus are overviewed (derivation formula, Daletskii-Krein formula, Dynkin’s formula, Campbell-Hausdorff theorem and others).
In Chapter II the method of ordered representation is introduced at first by examples, and then in a systematic way. For evaluation of the ordered representation operators the Yang-Baxter equation, the Jacobi condition, the Poincaré-Birkhoff-Witt theorem are used. The method is shown for the representations of Lie groups and functions of their generators.
In Chapter III the method developed in the previous chapter is applied to differential equations. Two kinds of problems are considered in this chapter:
a) find the inverse of a differential operator \(P\); b) find the resolvent operator for a Cauchy problem.
The method of ordered representation is applied for analysis of the following types of equations: difference-differential equations; electromagnetic waves in plasma equations; equations with symbols growing in infinity; geostrophic wind equations, and degenerate equations.
In Chapter IV the functional-analytic background of noncommutative analysis (convergence, symbol spaces and generators, functions of operators in scales of spaces) is discussed.
In Appendix A representation of Lie algebras and Lie groups is given. In Appendix B pseudodifferential operators are described.

47A60 Functional calculus for linear operators
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
47-XX Operator theory
44-XX Integral transforms, operational calculus
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory