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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).

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 |