An operator-theoretic formulation of asynchronous exponential growth. (English) Zbl 0654.47021

A strongly continuous semigroup of bounded operators \(\{\) T(t),t\(\geq 0\}\) in a Banach space X is said to have asynchronous exponential growth with intrinsic growth constant \(\lambda_ 0\in {\mathbb{R}}\) if there is a finite range operator \(P_ 0\) in X such that \[ \lim_{t\to \infty}e^{- \lambda_ 0t}T(t)=P_ 0. \] The author aims to establish necessary and sufficient conditions for \(\{\) T(t),t\(\geq 0\}\) to have asynchronous exponential growth. He first proves a necessary and sufficient condition in terms of the spectral properties of the generator of the semigroup. Then he studies how to guarantee that such properties are fulfilled. To this end he considers the case in which X is a Banach lattice and makes use of results of the Tübingen school. Finally, he applies his result to models describing cell population growth.
Reviewer: P.de Mottoni


47D03 Groups and semigroups of linear operators
92D25 Population dynamics (general)
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