The differentiation operator on discrete function spaces of a tree

TL;DR

This paper investigates the properties of the differentiation operator on discrete function spaces over infinite rooted trees, including boundedness, compactness, spectrum, and isometry conditions, with applications to various specialized spaces.

Contribution

It provides a comprehensive analysis of the differentiation operator on tree-based function spaces, connecting it with composition operators and exploring its spectral and norm properties.

Findings

Boundedness and compactness conditions established

Spectrum and isometry conditions characterized

Applications to Lipschitz, Banach, and Hardy spaces included

Abstract

In this paper, we study the differentiation operator acting on discrete function spaces; that is spaces of functions defined on an infinite rooted tree. We discuss, through its connection with composition operators, the boundedness and compactness of this operator. In addition, we discuss the operator norm and spectrum, and consider when such an operator can be an isometry. We then apply these results to the operator acting on the discrete Lipschitz space and weighted Banach spaces, as well as the Hardy spaces defined on homogeneous trees.