Transition operators on co-compact $G$-spaces

Saloff‐Coste, Laurent; Woess, Wolfgang

doi:10.4171/rmi/473

Cited by 15 publications

(24 citation statements)

References 31 publications

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“…The functions F (−1,n) , n ∈ N, of Corollary 4.16(b) are nondifferentiable at the vertices, and the functions G e 0,n , n ∈ N 0 , of Proposition 3.6 are even discontinuous, whence they have to be expressed as Fourier series in terms of the functions F nN/2 , n ∈ Z. The last formula for the spectral radius of A was first found by Saloff-Coste and Woess [16] by a completely different method, and indeed, the latter was the starting point for the present investigation. A closer look at spec(A) may be of interest.…”

Section: Final Remarks and Examplesmentioning

confidence: 93%

The spectrum of the averaging operator on a network (metric graph)

Cartwright¹,

Woess²

2007

Illinois J. Math.

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A network is a countable, connected graph X viewed as a one-complex, where each edge [x, y] = [y, x] (x, y ∈ X 0 , the vertex set) is a copy of the unit interval within the graph's one-skeleton X 1 and is assigned a positive conductance c(xy). A reference "Lebesgue" measure on X 1 is built up by using Lebesgue measure with total mass c(xy) on each edge [x, y]. There are three natural operators on X: the transition operator P acting on functions on X 0 (the reversible Markov chain associated with c), the averaging operator A over spheres of radius 1 on X 1 , and the Laplace operator ∆ on X 1 (with Kirchhoff conditions weighted by c at the vertices). The relation between the ℓ 2 -spectrum of P and the H 2 -spectrum of ∆ was described by Cattaneo [4]. In this paper we describe the relation between the ℓ 2 -spectrum of P and the L 2 -spectrum of A.(2.15) Lemma. Let −1 < λ < 1. Then the operator J λ = JS + λ J is compact and self-adjoint on the Hilbert space L 2 λ .Proof. If u, v ∈ L 2 [0, 1], then using

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Section: Final Remarks and Examplesmentioning

confidence: 93%

The spectrum of the averaging operator on a network (metric graph)

Cartwright¹,

Woess²

2007

Illinois J. Math.

Self Cite

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“…This theorem is a slight generalization of the following recent result of Saloff-Coste and Woess [26] The case where G is transitive has been treated in [22] and the case where M is the universal cover of a compact manifold has been proved in [1].…”

Section: Theorem 2 (See Theorem 114 and Corollary 1114) The Followmentioning

confidence: 93%

“…This provides a general and direct explanation for a phenomenon that has been pointed out in particular cases 1 [20,1,27,29,22,26].…”

Section: Introductionmentioning

confidence: 99%

Large-scale Sobolev inequalities on metric measure spaces and applications

Tessera¹

2008

Rev. Mat. Iberoam.

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For functions on a metric measure space, we introduce a notion of "gradient at a given scale". This allows us to define Sobolev inequalities at a given scale. We prove that satisfying a Sobolev inequality at a large enough scale is invariant under large-scale equivalence, a metric-measure version of coarse equivalence. We prove that for a Riemmanian manifold satisfying a local Poincaré inequality, our notion of Sobolev inequalities at large scale is equivalent to its classical version. These notions provide a natural and efficient point of view to study the relations between the large time on-diagonal behavior of random walks and the isoperimetry of the space. Specializing our main result to locally compact groups, we obtain that the L p -isoperimetric profile, for every 1 ≤ p ≤ ∞ is invariant under quasiisometry between amenable unimodular compactly generated locally compact groups. A qualitative application of this new approach is a very general characterization of the existence of a spectral gap on a quasi-transitive measure space X, providing a natural point of view to understand this phenomenon.

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“…it is an unbounded self-adjoint operator in L 2 (X 1 , m 1 ), see Theorem 4 below. Finally, the third operator A is the averaging operator over unit balls introduced in [12] in generalizing the notion of a transition operator [38]; it acts as…”

Section: Introductionmentioning

confidence: 99%

New Relations Between Discrete and Continuous Transition Operators on (Metric) Graphs

Lenz

Pankrashkin

2015

Integr. Equ. Oper. Theory

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We establish several new relations between the discrete transition operator, the continuous Laplacian and the averaging operator associated with combinatorial and metric graphs. It is shown that these operators can be expressed through each other using explicit expressions. In particular, we show that the averaging operator is closely related with the solutions of the associated wave equation. The machinery used allows one to study a class of infinite graphs without assumption on the local finiteness.

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Transition operators on co-compact $G$-spaces

Cited by 15 publications

References 31 publications

The spectrum of the averaging operator on a network (metric graph)

The spectrum of the averaging operator on a network (metric graph)

Large-scale Sobolev inequalities on metric measure spaces and applications

New Relations Between Discrete and Continuous Transition Operators on (Metric) Graphs

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