Multiplication operators between Lipschitz-type spaces on a tree

Abstract

Let $\mathcal{L}$ be the space of complex-valued functions $f$ on the set of vertices $T$ of an rooted infinite tree rooted at $o$ such that the difference of the values of $f$ at neighboring vertices remains bounded throughout the tree, and let ${\mathcal{L}}_{\text{w}}$ be the set of functions $f\in \mathcal{L}$ such that $|f(v)-f(v^{-})|=O(|v{|}^{-1})$, where $|v|$ is the distance between $o$ and $v$ and $v^{-}$ is the neighbor of $v$ closest to $o$. In this article, we characterize the bounded and the compact multiplication operators between $\mathcal{L}$ and ${\mathcal{L}}_{\text{w}}$, and provide operator norm and essential norm estimates. Furthermore, we characterize the bounded and compact multiplication operators between ${\mathcal{L}}_{\text{w}}$ and the space $L^{\infty }$ of bounded functions on $T$ and determine their operator norm and their essential norm. We establish that there are no isometries among the multiplication operators between these spaces.

References in corpus (1)

• Multiplication operators on the weighted Lipschitz space of a tree

Cited by in corpus (5)

• Multiplication operators on the iterated logarithmic Lipschitz spaces of a tree
• Discrete Analogue of Generalized Hardy Spaces and Multiplication Operators on Homogenous Trees
• Multiplication operators between iterated logarithmic Lipschitz spaces of a tree
• Multiplication operators between discrete Hardy spaces on rooted trees
• Composition operators on Hardy spaces of the homogenous rooted trees