On the Traces of Operators (from Grothendieck to Lidskii)

[摘要] In this paper, the reader’s attention is drawn to some notionsthat are classical in linear algebra but become more subtle todeal with in the context of infinite-dimensional vector spaces(endowed with a norm or a vector space topology). The Fredholm theory of integral equations, which will be mentionedat the end of the article, shares many common points withthe systems of linear equations taught in the first year of undergraduate studies, except that the endomorphisms operateon Banach spaces of the form C(K) (the space of continuousfunctions on a compact K) or Lp(Ω) for a given measure onΩ. In Fredholm theory, the notions of trace and determinant,as well as their relations to eigenvalues, of course play thesame role as in finite dimension. Fredholm’s seminal article“Sur une classe d’équations fonctionnelles” [Acta Mathematica, 27, pp. 365–390] goes back to 1903, a time when functional analysis was poorly developed. It was the starting pointof many works that partially motivated the advances in thefield throughout the 20th century (Hilbert, Banach, Fréchet,Dieudonné, Schwartz, Grothendieck, Sobolev, Gelfand, Kreinand many others).