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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| Format: | Article |
| Language: | English |
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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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| author | Robert F. Allen Flavia Colonna Glenn R. Easley |
| author_facet | Robert F. Allen Flavia Colonna Glenn R. Easley |
| author_sort | Robert F. Allen |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-abe72966bbd1467d8bf9392f46645c42 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-abe72966bbd1467d8bf9392f46645c422025-02-03T05:50:11ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252011-01-01201110.1155/2011/472495472495Multiplication Operators between Lipschitz-Type Spaces on a TreeRobert F. Allen0Flavia Colonna1Glenn R. Easley2Department of Mathematics, University of Wisconsin-La Crosse, La Crosse, WI 54601, USADepartment of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USASystem Planning Corporation, Arlington, VA 22209, USALet 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.http://dx.doi.org/10.1155/2011/472495 |
| spellingShingle | Robert F. Allen Flavia Colonna Glenn R. Easley Multiplication Operators between Lipschitz-Type Spaces on a Tree International Journal of Mathematics and Mathematical Sciences |
| title | Multiplication Operators between Lipschitz-Type Spaces on a Tree |
| title_full | Multiplication Operators between Lipschitz-Type Spaces on a Tree |
| title_fullStr | Multiplication Operators between Lipschitz-Type Spaces on a Tree |
| title_full_unstemmed | Multiplication Operators between Lipschitz-Type Spaces on a Tree |
| title_short | Multiplication Operators between Lipschitz-Type Spaces on a Tree |
| title_sort | multiplication operators between lipschitz type spaces on a tree |
| url | http://dx.doi.org/10.1155/2011/472495 |
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