Discussion of methods in theory of Partial Differential Operators (and Pseudo-Differential Operators) which are based on Gel'fand theory of commutative Banach Algebras. These methods all are discussed for the (non-compact) manifold Rn, for Lp/-SobolefF spaces Wp,k(including k = #x221E;) and for operators of "Laplace-type", which, crudely, amounts to "elliptic type". Generalizations to general non-compact manifolds and to hypo-elliptic operators have been worked out in detail in 6, for example. Two types of symbol are introduced for a (Pseudo)-differential operator: the Laplace-symbol-quotients(of order s) and the symbol #x3C3;. Among the results we have (a) necessary and sufficient criteria for the existence of an W:p,-Fredholm inverse, (b) essential self-adjointness and (c) characterization of the essential spectrum of elliptic (Pseudo-) Differential operators.
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