Illinois Journal of Mathematics

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 $[x,y]=[y,x]$ ($x,y\in X^0$, the vertex set) is a copy of the unit interval within the graph's one-skeleton $X^1$ and is assigned a positive conductance $\mathsf{c}(xy)$. A reference "Lebesgue" measure $X^1$ is built up by using Lebesgue measure with total mass $\mathsf{c}(xy)$ on each edge $xy$. There are three natural operators on $X$: the transition operator $P$ acting on functions on $X^0$ (the reversible Markov chain associated with $\mathsf{c}$), the averaging operator $A$ over spheres of radius~1 on $X^1$, and the Laplace operator $\Delta$ on $X^1$ (with Kirchhoff conditions weighted by $\mathsf{c}$ at the vertices). The relation between the $\ell^2$-spectrum of $P$ and the $H^2$-spectrum of~$\Delta$ was described by {Cattaneo} \cite{Cat}. In this note we describe the relation between the $\ell^2$-spectrum of $P$ and the $L^2$-spectrum of $A$.

Article information

Source
Illinois J. Math., Volume 51, Number 3 (2007), 805-830.

Dates
First available in Project Euclid: 13 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258131103

Digital Object Identifier
doi:10.1215/ijm/1258131103

Mathematical Reviews number (MathSciNet)
MR2379723

Zentralblatt MATH identifier
1152.47004

Citation

Cartwright, Donald I.; Woess, Wolfgang. The spectrum of the averaging operator on a network (metric graph). Illinois J. Math. 51 (2007), no. 3, 805--830. doi:10.1215/ijm/1258131103. https://projecteuclid.org/euclid.ijm/1258131103