The homotopy significant spectrum compared to the Krasnoselskii spectrum

TL;DR

This paper compares two generalizations of eigenvalues for homogeneous functionals, showing their relationships, proposing a modification for stability, and linking the Cheeger constant to the Krasnoselskii eigenvalue.

Contribution

It analyzes the relationship between the homotopy significant spectrum and the Krasnoselskii spectrum, introduces a stability-preserving modification, and connects the Cheeger constant to the second Krasnoselskii eigenvalue.

Findings

Krasnoselskii spectrum is contained in the homotopy significant spectrum in finite dimensions

Counterexample shows the reverse inclusion does not hold

Modified homotopy significant spectrum achieves stability

Abstract

How to generalize the concept of eigenvalues of quadratic forms to eigenvalues of arbitrary, even, homogeneous continuous functionals, if stability of the set of eigenvalues under small perturbations is required? We compare two possible generalizations, Gromov's homotopy significant spectrum and the Krasnoselskii spectrum. We show that in the finite dimensional case, the Krasnoselskii spectrum is contained in the homotopy significant spectrum, but provide a counterexample to the opposite inclusion. Moreover, we propose a small modification of the definition of the homotopy significant spectrum for which we can prove stability. Finally, we show that the Cheeger constant of a closed Riemannian manifold corresponds to the second Krasnoselskii eigenvalue.