Pseudodifferential Operators

The study of pseudodifferential operators emerged in the 1960’s, having its origins in the study of singular integro-differential operators. In fact, Friedrichs and Lax coined the term “pseudodifferential operator” in their 1965 paper entitled “Boundary Value Problems for First Order Operators”. Since that time, pseudodifferential operators have proved useful in many arenas of modern analysis and mathematical physics. They are particularly important to the study of elliptic equations and in the index theory for elliptic operators. Pseudodifferential operators “allow us not only to establish new theorems but also to have a fresh look at old ones and thereby obtain simpler and more transparent formulations of already known facts [3].” The aim of this paper will be to give an overview of pseudodifferential operators. We will motivate and define them and develop several of their important properties. Then we will define elliptic pseudodifferential operators and give several applications of them. In large part, this paper will follow the treatment in [2], with the most notable digression concerning the spectrum of elliptic operators.