An Inverse Theorem for the Perron--Frobenius Theorem
Shunsuke Tomioka
TL;DR
This work addresses the inverse Perron-Frobenius problem in infinite dimensions by constructing a cone P(u0) with axis along an eigenvector of A and proving that a bounded positive self-adjoint A with a simple maximal eigenvalue is positivity-improving relative to P(u0). It additionally establishes stability of positivity-improvement and ergodicity under small perturbations of the axis vector when a spectral gap is present. The authors apply the results to heat semigroups, showing positivity-improvement for small times and under perturbations, and illustrate with the magnetic Schrödinger operator. The findings provide a geometric mechanism for establishing positivity and uniqueness of ground states in spectral problems, with potential implications for spectral theory and quantum dynamics.
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 Schrödinger operator