Triple Variational Principles for Self-Adjoint Operator Functions

TL;DR

This paper develops triple variational principles to estimate eigenvalues of unbounded self-adjoint operator functions within spectral gaps, providing conditions for resolvent set inclusion and equality cases for norm resolvent continuous functions.

Contribution

It introduces a novel triple variational framework for eigenvalue bounds of unbounded self-adjoint operator functions, extending existing spectral theory methods.

Findings

Upper bounds for eigenvalues in spectral gaps

Conditions for points to be in the resolvent set

Equality cases for norm resolvent continuous functions

Abstract

For a very general class of unbounded self-adjoint operator function we prove upper bounds for eigenvalues which lie within arbitrary gaps of the essential spectrum. These upper bounds are given by triple variations. Furthermore, we find conditions which imply that a point is in the resolvent set. For norm resolvent continuous operator functions we show that the variational inequality becomes an equality.