Individual and Uniform Eventual Positivity of Self-Adjoint Semigroups

Abstract

We construct a real self-adjoint semigroup on an $\ell^2$-space which is individually eventually strongly positive with respect to a strictly positive vector, but whose operators fail to be positive at arbitrarily large times. Moreover, the generator of this semigroup has compact resolvent. Consequently, individual and uniform eventual positivity are not equivalent for self-adjoint semigroups on $L^2$-spaces, even under the additional assumption that the generator has compact resolvent. The construction consists of a positive self-adjoint rank-one perturbation of a diagonal operator and a sequence of small spectral shifts on antisymmetric two-point modes.