Abstract
Let be the space of complex-valued functions on the set of vertices of an infinite tree rooted at such that the difference of the values of at neighboring vertices remains bounded throughout the tree, and let be the set of functions such that , where is the distance between and and is the neighbor of closest to . In this paper, we characterize the bounded and the compact multiplication operators between and and provide operator norm and essential norm estimates. Furthermore, we characterize the bounded and compact multiplication operators between and the space of bounded functions on and determine their operator norm and their essential norm. We establish that there are no isometries among the multiplication operators between these spaces.
Highlights
Let X and Y be complex Banach spaces of functions defined on a set Ω
The Bloch space can be described as the set consisting of the Lipschitz functions between metric spaces from endowed with the Poincaredistance ρ to endowed with the Euclidean distance, a fact that was proved by the second author in 1 see 2
In 5, the last two authors defined the Lipschitz space L on a tree T as the set consisting of the functions f : T → which are Lipschitz with respect to the distance d on T and the Euclidean distance on
Summary
Let X and Y be complex Banach spaces of functions defined on a set Ω. They have a boundary, which is defined as the set of equivalence classes of paths which differ by finitely many vertices. In 5 , the last two authors defined the Lipschitz space L on a tree T as the set consisting of the functions f : T → which are Lipschitz with respect to the distance d on T and the Euclidean distance on. In 6 , we introduced the weighted Lipschitz space on a tree T as the set Lw of the functions f : T → such that sup|v|Df v < ∞ The interest in this space is due to its connection to the bounded multiplication operators on L. The multiplication operators between L and L∞ were studied by the last two authors in 7
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