Perturbation of analytic operators and temporal regularity of discrete heat kernels

Blunck, Sönke. "Perturbation of analytic operators and temporal regularity of discrete heat kernels." Colloquium Mathematicae 86.2 (2000): 189-201. <http://eudml.org/doc/210849>.

@article{Blunck2000,
abstract = {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.},
author = {Blunck, Sönke},
journal = {Colloquium Mathematicae},
keywords = {analyticity; continuous time semigroup; powerbounded and analytic operators; perturbation; temporal regularity of random walks; graphs of polynomial volume growth},
language = {eng},
number = {2},
pages = {189-201},
title = {Perturbation of analytic operators and temporal regularity of discrete heat kernels},
url = {http://eudml.org/doc/210849},
volume = {86},
year = {2000},
}

TY - JOUR
AU - Blunck, Sönke
TI - Perturbation of analytic operators and temporal regularity of discrete heat kernels
JO - Colloquium Mathematicae
PY - 2000
VL - 86
IS - 2
SP - 189
EP - 201
AB - 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.
LA - eng
KW - analyticity; continuous time semigroup; powerbounded and analytic operators; perturbation; temporal regularity of random walks; graphs of polynomial volume growth
UR - http://eudml.org/doc/210849
ER -