Year: 2024
Type: article
Abstract: We consider generators of positive $$C_0$$ C 0 -semigroups and, more generally, resolvent positive operators A on ordered Banach spaces and seek for conditions ensuring the negativity of their spectral bound $$s(A)$$ s ( A ) . Our main result characterizes $$s(A) < 0$$ s ( A ) < 0 in terms of so-called small-gain conditions that describe the behaviour of Ax for positive vectors x. This is new even in case that the underlying space is an $$L^p$$ L p -space or a space of continuous functions. We also demonstrate that it becomes considerably easier to characterize the property $$s(A) < 0$$ s ( A ) < 0 if the cone of the underlying Banach space has non-empty interior or if the essential spectral bound of A is negative. To treat the latter case, we discuss a counterpart of a Krein–Rutman theorem for resolvent positive operators. When A is the generator of a positive $$C_0$$ C 0 -semigroup, our results can be interpreted as stability results for the semigroup, and as such, they complement similar results recently proved for the discrete-time case. In the same vein, we prove a Collatz–Wielandt type formula and a logarithmic formula for the spectral bound of generators of positive semigroups.
Source: Journal of Evolution Equations
Authors: Jochen Glück, Andrii Mironchenko
Institutions: University of Wuppertal, University of Bayreuth
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
Awards: G2374186204, G4345095581
OpenAlex ID: W4405544363