Complexified cones. Spectral gaps and variational principles

TL;DR

This paper extends the theory of complexified real cones, providing new conditions for contractions, spectral gap estimates, and a variational formula for the leading eigenvalue, broadening the understanding of spectral properties of complex operators.

Contribution

It generalizes contraction conditions and spectral gap estimates for complex cones to Banach spaces and introduces a variational formula akin to Collatz-Wielandt for complex operators.

Findings

Generalized contraction conditions for complex operators on Banach spaces.

Provided an improved estimate of spectral gaps for complex cone contractions.

Derived a variational formula for the leading eigenvalue of complex operators.

Abstract

We consider contractions of complexified real cones, as recently introduced by Rugh in [Rugh10]. Dubois [Dub09] gave optimal conditions to determine if a matrix contracts a canonical complex cone. First we generalize his results to the case of complex operators on a Banach space and give precise conditions for the contraction and an improved estimate of the size of the associated spectral gap. We then prove a variational formula for the leading eigenvalue similar to the Collatz-Wielandt formula for a real cone contraction. Morally, both cases boil down to the study of suitable collections of 2 by 2 matrices and their contraction properties on the Riemann sphere.