1. Multiplication operators between Lipschitz-type spaces on a tree
- Author
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Allen, Robert F., Colonna, Flavia, and Easley, Glenn R.
- Subjects
Mathematics - Functional Analysis ,primary: 47B38, secondary: 05C05 - 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 $\mathcal{L}_{\textbf{w}}$ 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 $\mathcal{L}_{\textbf{w}}$, and provide operator norm and essential norm estimates. Furthermore, we characterize the bounded and compact multiplication operators between $\mathcal{L}_{\textbf{w}}$ 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.
- Published
- 2022
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