An Inverse Theorem for the Perron--Frobenius Theorem

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