Multiplication Operators between Lipschitz-Type Spaces on a Tree
Let ℒ be the space of complex-valued functions 𝑓 on the set of vertices 𝑇 of an infinite tree rooted at 𝑜 such that the difference of the values of 𝑓 at neighboring vertices remains bounded throughout the tree, and let ℒ𝐰 be the set of functions 𝑓∈ℒ such that |𝑓(𝑣)−𝑓(𝑣−)|=𝑂(|𝑣|−1), where |𝑣| is the...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2011-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2011/472495 |
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| Summary: | Let ℒ be the space of complex-valued functions 𝑓 on the set of vertices 𝑇 of an infinite tree rooted at 𝑜 such that the difference of the values of 𝑓 at neighboring vertices remains bounded throughout the tree, and let ℒ𝐰 be the set of functions 𝑓∈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 paper, we characterize the bounded and the compact multiplication operators between ℒ and ℒ𝐰 and provide operator norm and essential norm estimates. Furthermore, we characterize the bounded and compact multiplication operators between ℒ𝐰 and the space 𝐿∞ 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. |
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| ISSN: | 0161-1712 1687-0425 |