The spectral theory and its applications. Generalities and compact operators

The aim of spectral theory is to elucidate the structure of linear operators so that they can be decomposed into a collection of elementary operators, thus simplifying the solution of the problems in which they are involved. In the case of matrices, or in other words in finite dimensions, algebraic methods involving polynomials can be used to arrive at Jordan's form, which translates the decomposition of the operator into the sum of multiplication operators and a nilpotent operator. The ideal case is that of symmetric or self-adjoint matrices, in which the nilpotent operator is necessarily zero, giving the matrix a diagonal structure in an eigenvector basis. An abundant and complex literature deals with the numerical aspects of spectral decomposition of large matrices, and bears witness to the fact that simple and well-known theoretical results are not necessarily easy to implement in practice (cf. the article calculating eigenvalues in the same collection).

A decisive step was taken when spectral theory was applied to the study of equations, whether integral or partial differential, in infinite-dimensional spaces. The first results, relating to the study of integral equations, were obtained by Fredholm and then Hilbert, and generalized by F. Riesz into a theory of compact operators. These results depend on tools derived from functional analysis, but are close in many respects to those of finite dimension, unlike Stone's generalization to non-compact self-adjoint operators, in which measure theory plays an essential role. An important part of the subsequent developments, relating to unbounded operators and operator algebras, results from von Neumann's work and was initiated under the impetus of quantum mechanics.

In this article, we only present a general presentation of bounded operators and some aspects of the spectral theory of compact operators.