Toeplitz operators on $\mathcal L^p$-spaces of a tree

TL;DR

This paper characterizes spectral properties, self-adjointness, positivity, and finite rank conditions of Toeplitz operators on p spaces over infinite trees, advancing understanding of operator theory in graph-based function spaces.

Contribution

It provides new characterizations and criteria for Toeplitz operators on p spaces over trees, including spectral, self-adjointness, positivity, and finite rank conditions.

Findings

Spectra of Toeplitz operators are characterized.

Conditions for self-adjointness and positivity are established.

A necessary and sufficient condition for finite rank is derived.

Abstract

Let $T$ be a rooted, countable infinite tree without terminal vertices. In the present paper, we characterize the spectra, self-adjointness and positivity of Toeplitz operators on the spaces of $p$-summable functions on $T$. Moreover, we obtain a necessary and sufficient condition for Toeplitz operators to have finite rank on such function spaces.