On Rational Approximations of Groups of Operators

Abstract

In an earlier paper (P. Brenner and V. Thomée, On rational approximations of semi-groups, SIAM J. Numer. Anal., 16 (1979) pp. 683–694) we discussed stability and convergence of rational approximations $r^n (hA)\,nh = t$, of a strongly continuous semigroup $e^{tA} $ on a Banach space X, when $|r(z) \leqq 1|$ for $\operatorname{Re} z \leqq 0$. In the present paper we show, with applications to hyperbolic problems in mind, that if A generates a group on X, our previous results extend to the case when $|r(z) \leqq 1|$ for $\operatorname{Re} z = 0$ and $e^{tA} $ is a bounded operator between two Banach spaces $X_0 $ and $X_1 $ with $X \cap X_0 $ dense in $X_0 $.