Composition Operators on the Lipschitz Space of a Tree
Composition Operators on the Lipschitz Space of a Tree
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树的 Lipschitz 空间上的复合算子
DOI：
10.1007/s00009-013-0308-7
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
G. Easley
中科院分区：
文献类型：
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作者：
R. F. Allen;Flavia Colonna;G. Easley
无限树的Lipschitz空间定义为由函数组成的空间，使得$$\beta_f = {\rm sup}\{|f（v）- f（v^-）|：v \in T\backslash\{o\}，\，v^- {\rm parent \，of \，} v\}$$是有限的。在范数下是一个Banach空间。本文刻划了Lipschitz空间上的诱导复合算子是有界的、紧的或等距的函数φ映射到T自身.具体地说，它表明，有界复合算子的符号是Lipschitz映射的Tinto本身视为一个度量空间下的边计数距离。诱导紧算子的符号具有有限值域，而诱导上等距的符号恰好是固定根的映射，且其相邻顶点的像重合或本身就是相邻顶点。最后，详细研究了等距算子的谱。
The Lipschitz spaceof an infinite treeTrooted atois defined as the space consisting of the functionssuch that $$\beta_f = {\rm sup}\{|f(v) - f(v^-)| : v \in T\backslash\{o\}, \,v^- {\rm parent \, of \,} v\}$$is finite. Under the normis a Banach space. In this article, the functionsφmappingTinto itself whose induced composition operatoron the Lipschitz space is bounded, compact, or an isometry, are characterized. Specifically, it is shown that the symbols of the bounded composition operators are the Lipschitz maps ofTinto itself viewed as a metric space under the edge-counting distance. The symbols inducing compact operators have finite range while those inducing isometries onare precisely the onto maps fixing the root and whose images of neighboring vertices coincide or are themselves neighboring vertices. Finally, the spectrum of the operatorsthat are isometries is studied in detail.