An Evaluation on Numerous Applications of Linear Differential Operators

Exploring the Analogy between Linear Differential Operators and Matrices

Abstract

At the Edinburgh congress 12 years ago Gârding gave ageneral survey of the theory of linear partial differential operators. I shalltake his lecture as my starting point and try to give some idea of the laterdevelopment. Naturally it is necessary to concentrate on a few topics andignore others which are as interesting. I shall not try to list the omissionsbut wish to specify the limitation to questions concerning the existence andstructure of solutions of differential equations with constant, C¥or analytic coefficients. In this paper we will begin to take a more sophisticatedapproach to differential equations. We will define, with some care, the notionof a linear differential operator, and explore the analogy between suchoperators and matrices.