Perturbation of analytic operators and temporal regularity of discrete heat kernels

In analogy to the analyticity condition ${\displaystyle \parallel A{e}^{tA}\parallel \le C{t}^{-1}}$, t > 0, for a continuous time semigroup ${\displaystyle {\left(e^{tA}\right)}_{t\ge 0}}$, a bounded operator T is called analytic if the discrete time semigroup ${\displaystyle {\left(T^n\right)}_{n\in \mathbb{N}}}$ satisfies ${\displaystyle \parallel (T-I)T^n\parallel \le C{n}^{-1}}$, n ∈ ℕ. We generalize O. Nevanlinna's characterization of powerbounded and analytic operators T to the following perturbation result: if S is a perturbation of T such that ${\displaystyle \parallel R({\lambda }_0,T)-R({\lambda }_0,S)\parallel }$ is small enough for some ${\displaystyle {\lambda }_0\in \varrho \left(T\right)\cap \varrho \left(S\right)}$, then the type ${\displaystyle \omega }$ of the semigroup ${\displaystyle \left(e^{t(S-I)}\right)}$ also controls the analyticity of S in the sense that ${\displaystyle \parallel (S-I)S^n\parallel \le C(\omega +n^{-1})e^{\omega n}}$, n ∈ ℕ. As an application we generalize and give a simple proof of a result by M. Christ on the temporal regularity of random walks T on graphs of polynomial volume growth. On arbitrary spaces Ω of at most exponential volume growth we obtain this regularity for any powerbounded and analytic operator T on ${\displaystyle L_2(\Omega )}$ with a heat kernel satisfying Gaussian upper bounds.