Continuous Minimax Theorems

Abstract

In matrix theory, there exist useful extremal characterizations of eigenvalues and their sums for Hermitian matrices (due to Ky Fan, Courant– Fischer–Weyl and Wielandt) and some consequences such as the majorisation assertion in Lidskii’s theorem. In this paper, we extend these results to the context of self-adjoint elements of finite von Neumann algebras, and their distribution and quantile functions. This work was motivated by a lemma in [1] that described such an extremal characterization of the distribution of a self-adjoint operator affiliated to a finite von Neumann algebra – suggesting a possible analogue of the Courant–Fischer–Weyl minimax theorem for Hermitian matrices, for a self-adjoint operator in a finite von Neumann algebra.