Multiplication operators on the weighted Lipschitz space of a tree

Abstract

We study the multiplication operators on the weighted Lipschitz space ${\mathcal{L}}_{\text{w}}$ consisting of the complex-valued functions $f$ on the set of vertices of an infinite tree $T$ rooted at $o$ such that ${\sup }_{v\ne o}|v||f(v)-f(v^{-})|<\infty$, where $|v|$ denotes the distance between $o$ and $v$ and $v^{-}$ is the neighbor of $v$ closest to $o$. For the multiplication operator, we characterize boundedness, compactness, provide estimates on the operator norm and the essential norm, and determine the spectrum. We prove that there are no isometric multiplication operators or isometric zero divisors on ${\mathcal{L}}_{\text{w}}$.

arXiv admin note: text overlap with arXiv:2207.12212

Cited by in corpus (6)

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