Hyper-positive definite functions II: A complete study of branching-Toeplitz operators

TL;DR

This paper provides a comprehensive analysis of branching-Toeplitz operators on homogeneous trees, generalizing classical Toeplitz operators, and explores their applications in constructing determinantal point processes with branching structures.

Contribution

It introduces and thoroughly studies a new class of branching-Toeplitz operators and matrices, extending classical Toeplitz theory to tree-indexed settings.

Findings

Characterization of branching-Toeplitz operators on homogeneous trees

Development of a theory for branching-Toeplitz matrices

Construction of determinantal point processes on trees

Abstract

We introduce and give a more or less complete study of a family of branching-Toeplitz operators on the Hilbert space $\ell^2(T_q)$ indexed by a rooted homogeneous tree $T_q$ of degree $q\ge 2$. The finite dimensional analogues of such operators form a very natural family of structured sparse matrices called branching-Toeplitz matrices and will also be investigated. The branching-Toeplitz operators/matrices in this paper should be viewed as natural generalizations of the standard Toeplitz operators/matrices. We will apply our results to construct a family of determinantal point processes on homogeneous trees which are branching-type strong stationary stochastic processes.