Abstract
This chapter discusses operators and their symbols. Hörrnander and Duistermaat introduced an important class of operators, the Fourier integral operators, with which they obtained many new results in the theory of linear partial differential equations. Their results depended on the notion of the symbol of such an operator and a functional calculus relating operators to their symbols. The class of Fourier integral operators is not linear and is not closed under composition in general. The chapter improves this calculus to a class of operators containing the Fourier integral operators, which is closed under linear combinations and composition. The usual letters are used for the usual Schwartz spaces on a Riemannian manifold. The chapter discusses the basic definitions of the symbol and related notions as well as establishes the essentials of the functional calculus. The completeness of the symbol is proved.
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