Positive definite radial kernels on homogeneous trees and products

Abstract

We give a new proof of a classical result which provides a one-to-one correspondence between positive definite radial kernels on a homogeneous tree and finite Borel measures on the interval $[-1,1]$. Our methods allow us to find a new characterisation in terms of positive trace-class operators on ${\ell }_2$. Furthermore, we extend both characterisations to finite products of homogeneous trees. The proof relies on a formula for the norm of radial Schur multipliers, in the spirit of Haagerup--Steenstrup--Szwarc, and a variation of the Hamburger moment problem.

v2: 35 pages, minor changes. Accepted for publication in Journal of Operator Theory

References in corpus (2)

• Schur Multipliers and Spherical Functions on Homogeneous Trees
• Radial Schur multipliers on some generalisations of trees

Cited by in corpus (1)

• Pointwise convergence of noncommutative Fourier series