PreprintCC BY 4.0arXiv
Individual and Uniform Eventual Positivity of Self-Adjoint Semigroups
Abstract
We construct a real self-adjoint semigroup on an ℓ2-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 L2-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.
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BibTeX
@misc{dobrick2026individual,
title = {{Individual and Uniform Eventual Positivity of Self-Adjoint Semigroups}},
author = {Alexander Dobrick},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00425},
eprint = {2609.00425},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00425},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Individual and Uniform Eventual Positivity of Self-Adjoint Semigroups
AU - Alexander Dobrick
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00425
UR - https://arxiv.org/abs/2609.00425
AB - We construct a real self-adjoint semigroup on an ℓ^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.
ER -