We show that the growth rates of solutions of the abstract differential equationsẋ(t) = Ax(t),ẋ(t) = A −1 x(t), and the difference equation x d (n + 1) = (A + I)(A − I) −1 x d (n) are closely related. Assuming that A generates an exponentially stable semigroup, we show that on a general Banach space the lowest growth rate of the semigroup (e A −1 t) t≥0 is O(4 √ t), and for ((A + I)(A − I) −1) n it is O(4 √ n). The similarity in growth holds for all Banach spaces. In particular, for Hilbert spaces the best estimates are O(log(t)) and O(log(n)), respectively. Furthermore, we give conditions on A such that the growth rate of ((A + I)(A − I) −1) n is O(1), i.e., the operator is power bounded.
Abstract. For semigroups and for bounded operators we introduce the new notion of Bergman distance. Systems with a finite Bergman distance share the same stability properties, and the Bergman distance is preserved under the Cayley transform. This way, we get stability results in continuous and discrete time. As an example, we show that bounded perturbations lead to pairs of semigroups with finite Bergman distance. This is extended to a class of Desch-Schappacher perturbations.Mathematics Subject Classification (2010). Primary 47D60; Secondary 93D05.
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