Jordan decomposition for differential operators

Abstract

One of the most well-known theorems of linear algebra states that every linear operator on a complex vector space has a Jordan decomposition. There are now numerous ways to prove this theorem, however a standard method of proof relies on the existence of an eigenvector. Given a finite-dimensional, complex vector space V, every linear operator T : V → V has an eigenvector (i.e. a v ∈ V such that (T − λI)v = 0 for some λ ∈ C). If we are lucky, V may have a basis consisting of eigenvectors of T, in which case, T is diagonalisable. Unfortunately this is not always the case. However, by relaxing the condition n (T − λI)v = 0 to the weaker condition (T − λI)nv = 0 for somen ∈ N, we can always obtain a basis of generalised eigenvectors. In fact, there is a canonical decomposition of V into generalised eigenspaces and this is essentially the Jordan decomposition.The topic of this thesis is an analogous theorem for differential operators. The existence of a Jordan decomposition in this setting was first proved by Turrittin following work of Hukuhara in the one-dimensional case. Subsequently, Levelt proved uniqueness and provided a more conceptual proof of the original result. As a corollary, Levelt showed that every differential operator has an eigenvector. He also noted that this was a strange chain of logic: in the linear setting, the existence of an eigenvector is a much easier result and is in fact used to obtain the Jordan decomposition. Levelt remarked that a direct proof of his corollary would provide a much simpler proof of the Jordan decomposition for differential operators. It is this remark that stimulated the work of this thesis. Although there have been numerous alternative proofs and applications of the Hukuhara-Levelt-Turrittin theorem, it appears that Levelt's suggestion has not been carried out in the literature. Our goal is to provide a proof of the Hukuhara-Levelt-Turrittin theorem that mimics the proof of the usual Jordan decomposition theorem.