The Forward and Backward Shift on the Hardy Space of a Tree

TL;DR

This paper explores the properties of forward and backward shift operators on Hardy spaces defined on trees, including boundedness, norms, isometry conditions, spectra, and hypercyclicity, providing foundational insights into these operators in a graph-theoretic context.

Contribution

It introduces the study of shift operators on Hardy spaces of trees, characterizing their boundedness, norms, isometric conditions, spectra, and hypercyclicity for the first time.

Findings

Identified conditions for boundedness of shifts.

Computed norms of the shifts.

Characterized when shifts are isometries and hypercyclic.

Abstract

In this paper we initiate the study of the forward and backward shifts on the Hardy space of a tree and the little Hardy space of a tree. In particular, we investigate when these shifts are bounded, find the norm of the shifts if they are bounded, characterize the trees in which they are an isometry, compute the spectrum in some concrete examples, and completely determine when they are hypercyclic.