Asymptotics of resolvent iterates for abstract potentials (Q1882662)

An operator \(A\) is called an abstract potential if the resolvent set of \(A\) contains an open right half plane and if there is a constant \(a>0\) such that the function \((\text{Re}\cdot -a)(\cdot I-A)^{-1}\) is bounded on the set \(\mathbb C_a:=\{\lambda\in\mathbb C\mid \text{ Re}\lambda >a\}\). The main result of this paper is: Every abstract potential satisfies \(\| (\text{Re}\lambda -a)^n(\cdot I-A)^{-n}\| < Men\) for \(\lambda \in \mathbb C_a\) and every \(n\in \mathbb N\). Here \(M\) is the bound of \((\text{Re}\cdot -a)(\cdot I-A)^{-1}\) on \(\mathbb C_a\).