Discrete Analogue of Generalized Hardy Spaces and Multiplication Operators on Homogenous Trees

TL;DR

This paper introduces discrete Hardy spaces on homogeneous trees, explores their properties, and characterizes bounded, compact, and isometric multiplication operators within these spaces.

Contribution

It defines a new discrete analogue of generalized Hardy spaces on homogeneous trees and analyzes the properties and operator theory related to these spaces.

Findings

Spaces are complete and separable under certain conditions

Bounded and compactness criteria for multiplication operators are established

Spectral properties of multiplication operators are characterized

Abstract

In this article, we define discrete analogue of generalized Hardy spaces and its separable subspace on a homogenous rooted tree and study some of its properties such as completeness, inclusion relations with other spaces, separability, growth estimate for functions in these spaces and their consequences. Equivalent conditions for multiplication operators to be bounded and compact are also obtained. Furthermore, we discuss about point spectrum, approximate point spectrum and spectrum of multiplication operators and discuss when a multiplication operator is an isometry.