Perturbation of analytic operators and temporal regularity of discrete heat kernels

Sönke Blunck¹

¹

Colloquium Mathematicum, Tome 86 (2000) no. 2, pp. 189-201

Résumé

In analogy to the analyticity condition $∥ Ae^{tA}∥ ≤ Ct^{-1}$, t > 0, for a continuous time semigroup $(e^{tA})_{t ≥ 0}$, a bounded operator T is called analytic if the discrete time semigroup $(T^n)_{n ∈ ℕ}$ satisfies $∥ (T-I)T^{n}∥ ≤ Cn^{-1}$, 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 $∥ R(λ_0,T)-R(λ_0,S)∥$ is small enough for some $λ_{0} ∈ ϱ(T) ∩ ϱ(S)$, then the type $ω$ of the semigroup $(e^{t(S-I)})$ also controls the analyticity of S in the sense that $∥(S-I)S^{n}∥ ≤ C(ω+n^{-1})e^{ωn}$, 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 $L_{2}(Ω)$ with a heat kernel satisfying Gaussian upper bounds.

Détail

DOI : 10.4064/cm-86-2-189-201