# Perturbation of analytic operators and temporal regularity of discrete heat kernels

Colloquium Mathematicae (2000)

- Volume: 86, Issue: 2, page 189-201
- ISSN: 0010-1354

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topBlunck, 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 -

## References

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- [H-SC] W. Hebisch and L. Saloff-Coste, Gaussian estimates for Markov chains and random walks on graphs, Ann. Probab. 21 (1993), 673-709. Zbl0776.60086
- [J] O. D. Jones, Transition probabilities for the simple random walk on the Sierpiński graph, Stochastic Process. Appl. 61 (1996), 45-69. Zbl0853.60058
- [N1] O. Nevanlinna, Convergence of Iterations for Linear Equations, Birkhäuser, Basel, 1993.
- [N2] O. Nevanlinna, On the growth of the resolvent operators for power bounded operators, in: Linear Operators, J. Janas, F. H. Szafraniec and J. Zemánek (eds.), Banach Center Publ. 38, Inst. Math., Polish Acad. Sci., 1997, 247-264. Zbl0913.47004
- [P] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer, 1983. Zbl0516.47023

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