# Journal of Operator Theory

Volume 84, Issue 2, Fall 2020  pp. 487-519.

Positive definite radial kernels on homogeneous trees and products

Authors:  Ignacio Vergara
Author institution: Institute of Mathematics of the Polish Academy of Sciences, ul. Sniadeckich 8, 00-656 Warszawa, Poland

Summary: 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.

DOI: http://dx.doi.org/10.7900/jot.2019aug07.2243
Keywords: positive definite kernels, homogeneous trees, radial Schur multipliers, Hamburger moment problem