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

Abstract

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

Unitary dimension reduction for a class of self-adjoint extensions with applications to graph-like structures

“…We believe that, with some suitable modifications, similar relationships should exist for other type of operators, like the averaging operator [44] or the fourth order or mixed order operators appearing in the description of beams [45,20]. We hope to clarify the situation in subsequent works.…”

Dynamical systems associated with adjacency matrices

“…It is well-known that D −1 A is the transition matrix associated with a random walk on the graph and it has been shown in [CW07] that an interesting interplay exists between the spectral properties of D −1 A and those of a certain bounded, self-adjoint operator A over L 2 (G), where G is the quantum graph associated with G. Such an A can be interpreted as the operator that maps functions over a quantum graph G into their average over balls of radius 1 with respect to the canonical structure of a quantum graph as a metric measure space. Spectral relations between D −1 A and A as well as a representation of A in terms of the quantum graph Laplacian via self-adjoint functional calculus have been proved in [CW07,LP16]. The considerations in [CW07] suggest that the correct quantum graph counterpart of our (unnormalized!)…”

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

“…Furthermore, for all x ∈ X 0 and u, v ∼ x consider the functionals l xuv on dom S given by l xuv F = F(xv, 0) − F(xu, 0). By (12), all these functionals are continuous in the graph norm of S, hence their kernels are closed in the graph norm of S. As S is the restriction of S to the intersection of these kernels, it is a closed operator.…”

“…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…”