Multiplication operators on the iterated logarithmic Lipschitz spaces of a tree

TL;DR

This paper studies multiplication operators on a new class of iterated logarithmic Lipschitz spaces defined on infinite trees, providing characterizations of boundedness, compactness, spectrum, and isometric properties.

Contribution

It introduces the iterated logarithmic Lipschitz spaces on trees and characterizes key properties of multiplication operators acting on these spaces.

Findings

Boundedness and compactness criteria for multiplication operators.

Operator norm and essential norm estimates.

Spectral and isometric properties of multiplication operators.

Abstract

We introduce a class of iterated logarithmic Lipschitz spaces $\mathcal{L}^{(k)}$, $k\in\mathbb{N}$, on an infinite tree which arise naturally in the context of operator theory. We characterize boundedness and compactness of the multiplication operators on $\mathcal{L}^{(k)}$ and provide estimates on their operator norm and their essential norm. In addition, we determine the spectrum, characterize the multiplication operators that are bounded below, and prove that on such spaces there are no nontrivial isometric multiplication operators and no isometric zero divisors.