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Méthodes opératorielles. Traduit du russe par Djilali Embarek. (English) Zbl 0651.47039

Moscou: Éditions Mir. 707 p. (1987).
In this graduate text the author develops a new operational calculus: the calculus of ordered non-commutative operators, and proves the fundamental theorem of quasi-inversion for such operators. New algebraic and geometric notions tie together differential equations, spectral theory and symplectic geometry. The contents is as follows: introduction to operational calculus, abstract spaces, functional spaces, functions of regular operators, non-commutative operator calculus, T-product of hypoelliptic operators and spectral decomposition of a T-product, generalized Hamilton-Jacobi equation, canonical operator on a Lagrangean manifold with complex germs and proof of the fundamental theorem, and theory of linear equations on semi-modules.
The clear presentation is the result of three years of teaching on this subject by the author before 1973. One can find a relevant bibliography in a more recent work by the author and V. E. Nazaikinskii [Asimptoticheskie metody resheniya psevdodifferentsial’nykh uravnenii; English: Asymptotics of Operator and Pseudo-Differential Equations, Consultant Bureau, New York and London, 1988]. The reviewed book appeared in English [Operational Methods (1973; Zbl 0449.47002)] and in Spanish [Métodos operatorios (1981; Zbl 0528.47034)], the original is in Russian (1973; Zbl 0288.47042).
Reviewer: R.Vaillancourt

MSC:

47Gxx Integral, integro-differential, and pseudodifferential operators
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
58J40 Pseudodifferential and Fourier integral operators on manifolds
35S05 Pseudodifferential operators as generalizations of partial differential operators
35C20 Asymptotic expansions of solutions to PDEs
47F05 General theory of partial differential operators
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems