Positive irreducible semigroups and their long-time behaviour.

The notion usually refers to the interaction between three properties of operator semigroups: positivity, spectrum and long-time behaviour. These interactions gives rise to a profound theory with plenty of applications. By a brief walk-through of the field and with many examples, we highlight two aspects of the subject, both related to the long-time behaviour of semigroups: (i) The classical question how positivity of a semigroup can be used to prove convergence to an equilibrium as  → ∞. (ii) The more recent phenomenon that positivity itself sometimes occurs only for large , while being absent for smaller times. This article is part of the theme issue 'Semigroup applications everywhere'.