Beispiele von Pseudo-Differentialoperator-Algebrenfl

Abstract

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 L^p/-SobolefF spaces W_p,k (including k = ∞) 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-quotient ^s (of order s) and the symbol σ. 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.