A coordinate-free treatment of the minimax and maximini theorems for eigenvalues of self-adjoint operators on a Hilbert space

Abstract

The power and elegance of coordinate-free analysis is demonstrated by establishing the minimax and maximini theorems for eigenvalues of self-adjoint operators on a Hilbert space as trivial corollaries of a series of lemmas of astounding simplicity. Furthermore the analysis brings out the essential identity of the two results in contradistinction to what has been generally believed before. A host of other related results are also established and in particular the authors propose new simplified proofs of the inequalities of Weyl, Poincare and Bateman; the latter two of these are better known to physicists as the Hylleraas-Undheim and MacDonald theorems and are extensively used in establishing upper bound and convergence properties of energy eigenvalues of stationary states of quantum systems.