Radial Schur multipliers on some generalisations of trees

TL;DR

This paper characterizes radial Schur multipliers on products of trees and hyperbolic graphs, extending previous results and connecting conditions to trace class operators and Besov spaces.

Contribution

It extends the characterization of radial Schur multipliers to products of trees, hyperbolic graphs, and CAT(0) cube complexes, linking operator conditions to function spaces.

Findings

Characterization of radial Schur multipliers via trace class Hankel matrices.

Extension of results from trees to products of trees and hyperbolic graphs.

Provision of conditions for Schur multipliers on CAT(0) cube complexes.

Abstract

We give a characterisation of radial Schur multipliers on finite products of trees. The equivalent condition is that a certain generalised Hankel matrix involving the discrete derivatives of the radial function is a trace class operator. This extends Haagerup, Steenstrup and Szwarc's result for trees. The same condition can be expressed in terms of Besov spaces on the torus. We also prove a similar result for products of hyperbolic graphs and provide a sufficient condition for a function to define a radial Schur multiplier on a finite dimensional CAT(0) cube complex.