Stability criteria for positive semigroups on ordered Banach spaces

Titelangaben

Glück, Jochen ; Mironchenko, Andrii:
Stability criteria for positive semigroups on ordered Banach spaces.
In: Journal of Evolution Equations. Bd. 25 (2025) Heft 1 . - 12.
ISSN 1424-3202
DOI der Verlagsversion: https://doi.org/10.1007/s00028-024-01044-8

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Abstract

We consider generators of positive C₀-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). Our main result characterizes 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 $L^p$-space or a space of continuous functions. We also demonstrate that it becomes considerably easier to characterize the property 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 C₀-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.