Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6415954 | Linear Algebra and its Applications | 2016 | 23 Pages |
The result that goes essentially back to Euler [15] says that for any element a of a unital Banach algebra A with unit u, the limit limεâ0+â¡(u+εa)[εâ1t] (where [â ] denotes the integral part) exists for all tâR and equals eta. As developed by E. Hille [22, Thm. 12.2.1], in the case where a is replaced by the generator A of a strongly continuous semigroup {etA,tâ¥0} in a Banach space X, a proper counterpart of this formula is etA=limεâ0+â¡(IXâεA)â[εâ1t] strongly in X. Motivated by an example from mathematical biology (related to Rotenberg's model of cell growth [40]) we study convergence of a similar approximation in which u (resp. IX) is replaced by jâA (resp. JâL(X)) such that for some ââ¥2, jâ=u (resp. Jâ=IX).