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An intriguing feature of positive $C_0$-semigroups on function spaces (ormore generally on Banach lattices) is that their long-time behaviour is mucheasier to describe than it is for general semigroups. In particular, theconvergence of semigroup operators (strongly or in the operator norm) as timetends to infinity can be characterized by a set of simple spectral andcompactness conditions.  In the present paper, we show that similar theorems remain true for thelarger class of (uniformly) eventually positive semigroups - which recentlyarose in the study of various concrete differential equations.  A major step in one of our characterizations is to show a version of thefamous Niiro-Sawashima theorem for eventually positive operators. Severalproofs for positive operators and semigroups do not work in our setting anylonger, necessitating different arguments and giving our approach a distinctflavour.

Spectrum and convergence of eventually positive operator semigroups