On Hille-type approximation of degenerate semigroups of operators

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 limε→0+⁡(u+εa)[ε−1t] (where [⋅] denotes the integral part) exists for all t∈R 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,t≥0} in a Banach space X, a proper counterpart of this formula is etA=limε→0+⁡(IX−εA)−[ε−1t] 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 j∈A (resp. J∈L(X)) such that for some ℓ≥2, jℓ=u (resp. Jℓ=IX).