An Inverse Theorem for the Perron--Frobenius Theorem

TL;DR

This paper proves that for any bounded positive self-adjoint operator, there exists a Hilbert cone making it ergodic with a simple maximum eigenvalue, extending the Perron--Frobenius theorem to an inverse setting.

Contribution

It establishes an inverse Perron--Frobenius theorem by showing the existence of a Hilbert cone for any such operator, and analyzes a specific cone for this purpose.

Findings

Existence of a Hilbert cone making a given operator ergodic.

Construction of a specialized Hilbert cone for the inverse problem.

Application to the heat semigroup of the magnetic Schrödinger operator.

Abstract

The Perron--Frobenius theorem in infinite-dimensional Hilbert spaces can be breifly stated as follows: Given a Hilbert cone in a real Hilbert space, a bounded positive self-adjoint operator $A$ is ergodic with respect to this cone if and only if the maximum eigenvalue $\|A\|$ of $A$ is simple, and the corresponding eigenvector is strictly positive with respect to this cone. This paper addresses the inverse problem of the Perron--Frobenius theorem: Does there exist a Hilbert cone such that a given bounded positive self-adjoint operator $A$ becomes ergodic when its maximum eigenvalue $\|A\|$ is simple? We provide an affirmative answer to this question in this paper. Furthermore, we conduct a detailed analysis of a specialized Hilbert cone introduced to obtain this result. Additionally, we provide an illustrative example of an application of the obtained results to the heat semigroup generated by the magnetic Schr\"odinger operator