Methods of Noncommutative Analysis. Theory and Applications
Vladimir E. Nazaikinskii, Victor E. Shatalov, Boris Yu. Sternin

Methods of Noncommutative Analysis. Theory and Applications
(in series ,,de Gruyter Studies in Mathematics", vol. 11)
Berlin; New York: de Gruyter | 1996 | 373 pages | English | ISBN: 3110146320 | zip/djvu | 3MB
Provides a systematic and self-contained introduction to the basics of noncommutative analysis with interesting applications, ranging from classical to modern ones.

Noncommutative analysis, that is, the calculus of noncommuting operators, is one of the main tools in contemporary mathematics. Indeed, the theory of differential and pseudodifferential operators, various problems of algebra, functional analysis, and theoretical physics deal with functions of noncommuting operators. Noncommutative analysis occurs to be useful in such mathematical fields as the theory of geometric and asymptotic quantization, representation theory, the theory of quantum groups, and so on. In this book the reader will find a lot of examples from functional analysis, algebra, representation theory, and the theory of differential equations where non commutative analysis is involved.

Unfortunately, up to now there does not exist a sufficiently simple exposition of noncommutative analysis which might serve as an introduction to the subject for scientists who are just beginning to get acquainted with this area and that is why this book has been written. It is primarily addressed to those who are not specialists in noncommutative analysis.

At the same time, even the experienced mathematician can find in this book many new and interesting topics. Noncommutative analysis gives a new outlook both on quite traditional and modem mathematical topics such as representation theory, operator theory, the theory of (pseudo)differential operators, Yang-Baxter equations and others.

Table of Contents
I Elementary Notions of Noncommutative Analysis 1 Some Situations where Functions of Noncommuting Operators Arise
2 Functions of Noncommuting Operators: the Construction and Main Properties
3 Noncommutative Differential Calculus.
4 The Campbell-Hausdorff Theorem and Dynkin's Formula
5 Summary: Rules of "Operator Arithmetic" and Some Standard Techniques
II Method of Ordered Representation 1 Ordered Representation: Definition and Main Property
2 Some Examples
3 Evaluation of the Ordered Representation Operators
4 The Jacobi Condition and Poincar6-Birkhoff-Witt Theorem
5 The Ordered Representations, Jacobi Condition, and the Yang-Baxter
6 Representations of Lie Groups and Functions of Their Generators III Noncommutative Analysis and Differential Equations 1 Preliminaries
2 Difference and Difference-Differential Equations
3 Propagation of Electromagnetic Waves in Plasma
4 Equations with Symbols Growing at Infinity
5 Geostrophic Wind Equations
6 Degenerate Equations
7 Microlocal Asymptotic Solutions for an Operator with Double Characteristics IV Functional-Analytic Background of Noncommutative Analysis 1 Topics on Convergence
2 Symbol Spaces and Generators
3 Functions of Operators in Scales of Spaces Appendix A. Representation of Lie Algebras and Lie Groups
Appendix B. Pseudodifferential Operators
Glossary
Bibliographical Remarks
Bibliography
Index