Asymptotic properties of a positive semigroup, as \(t \to + \infty\) (Q1909792)

This paper is a continuation of two previous works by the same author in [Mat. Zametki 56, No. 2, 56-81 (Russian) (1994; Zbl 0843.35124) and ibid. 57, No. 1, 72-89 (Russian) (1995; Zbl 0840.45001)]. It uses notations, definitions and equations from the previous parts. The author studies the asymptotic behavior of a positive semigroup \(T(t)\), \(t\geq 0\), for \(t\to+\infty\) under certain conditions imposed on it. His approach is based on the representation \(T(t)= \sum^{N_\varepsilon}_{j= 0} \exp(A_jt)P_j+ T(t)P_\varepsilon\), where \(P_j\) are projections on certain eigenspaces, \(P_\varepsilon= I-\sum^{N_\varepsilon}_{j= 0} P_j\) and there is an appropriate estimate for \(|T(t)P_\varepsilon|\).