Radial Schur multipliers on some generalisations of trees

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.

Positive definite radial kernels on homogeneous trees and products

“…A characterisation of multi-radial multipliers on products of trees was given in [19], together with some estimates on the norms. We shall revisit the proof of this result and obtain an exact formula for the norm in the homogeneous case.…”

“…Observe that one of the inequalities is somehow implicit in [19]. By [19, Lemma 2.13], there exists a trace-class operator…”

“…Observe that the function (m, n) → (−1) |m|+|n| defines a Schur multiplier on X of norm 1 (see e.g. [19,Lemma 2.3]). Therefore…”

“…So again, this is a very different situation than the one of radial Schur multipliers. Indeed, the following is a consequence of [19, Theorem A] and [19,Proposition 7.5]. Theorem 1.6 ([19]).…”