Spectrum and convergence of eventually positive operator semigroups

Abstract

An intriguing feature of positive $C_0$-semigroups on function spaces (or more generally on Banach lattices) is that their long-time behaviour is much easier to describe than it is for general semigroups. In particular, the convergence of semigroup operators (strongly or in the operator norm) as time tends to infinity can be characterized by a set of simple spectral and compactness conditions. In the present paper, we show that similar theorems remain true for the larger class of (uniformly) eventually positive semigroups - which recently arose in the study of various concrete differential equations. A major step in one of our characterizations is to show a version of the famous Niiro-Sawashima theorem for eventually positive operators. Several proofs for positive operators and semigroups do not work in our setting any longer, necessitating different arguments and giving our approach a distinct flavour.

17 pages. This is version 3. Minor changes compared to v2; two additional references have been introduced

References in corpus (2)

• The Bi-Laplacian with Wentzell boundary conditions on Lipschitz domains
• Locally Eventually Positive Operator Semigroups

Cited by in corpus (7)

• Locally Eventually Positive Operator Semigroups
• Stability of (eventually) positive semigroups on spaces of continuous functions
• An operator theoretic approach to uniform (anti-)maximum principles
• Spectral properties of locally eventually positive operator semigroups
• Local uniform convergence and eventual positivity of solutions to biharmonic heat equations
• Domination of semigroups generated by regular forms
• Eventually positive semigroups: spectral and asymptotic analysis