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.