Abstract
A strongly continuous semigroup of bounded linear operators T ( t ) T(t) , t ⩾ 0 t \geqslant 0 , in the Banach space X X has asynchronous exponential growth with intrinsic growth constant λ 0 {\lambda _0} provided that there is a nonzero finite rank operator P 0 {P_0} in X X such that lim t → ∞ e − λ 0 t T ( t ) = P 0 {\lim _{t \to \infty }}{e^{ - {\lambda _0}t}}T(t) = {P_0} . Necessary and sufficient conditions are established for T ( t ) T(t) , t ⩾ 0 t \geqslant 0 , to have asynchronous exponential growth. Applications are made to a maturity-time model of cell population growth and a transition probability model of cell population growth.
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