A new version of the Gelfand-Hille theorem

Abstract

Let X be a complex Banach space and A∈L(X) with σ(A)={1}. We prove that for a vector x∈X, if ‖(Ak+A−k)x‖=O(kN) as k→+∞ for some positive integer N, then (A−I)N+1x=0 when N is even and (A−I)N+2x=0 when N is odd. This could be seemed as a new version of the Gelfand-Hille theorem. As a corollary, we also obtain that for a quasinilpotent operator Q∈L(X) and a vector x∈X, if ‖cos⁡(kQ)x‖=O(kN) as k→+∞ for some positive integer N, then QN+1x=0 when N is even and QN+2x=0 when N is odd. Moreover, if A satisfies the uniqueness property for the local resolvent, the condition σ(A)={1} could be replaced by and σx(A)={1}, which is a more local condition.