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The orbit of a bounded operator under the Möbius group modulo similarity equivalence
Israel Journal of Mathematics ( IF 1 ) Pub Date : 2020-05-20 , DOI: 10.1007/s11856-020-2016-x
Soumitra Ghara

Let Möb denote the group of biholomorphic automorphisms of the unit disc and (Möb · T ) be the orbit of a Hilbert space operator T under the action of Möb. If the quotient , where is the similarity between two operators is a singleton, then the operator T is said to be weakly homogeneous. In this paper, we obtain a criterion to determine if the operator M z of multiplication by the coordinate function z on a reproducing kernel Hilbert space is weakly homogeneous. We use this to show that there exists a Möbius bounded weakly homogeneous operator which is not similar to any homogeneous operator, answering a question of Bagchi and Misra in the negative. Some necessary conditions for the Möbius boundedness of a weighted shift are also obtained. As a consequence, it is shown that the Dirichlet shift is not Möbius bounded.

中文翻译:

莫比乌斯群模相似等价下有界算子的轨道

令Möb 表示单位圆盘的双全纯自同构群,(Möb·T) 是希尔伯特空间算符T 在Möb 作用下的轨道。如果两个算子之间的相似性的商 是单例,则称算子 T 是弱齐次的。在本文中,我们获得了一个判据来确定在再现核希尔伯特空间上乘以坐标函数 z 的算子 M z 是否是弱齐次的。我们用它来证明存在一个与任何齐次算子都不相似的莫比乌斯有界弱齐次算子,否定地回答了 Bagchi 和 Misra 的问题。还获得了加权偏移的莫比乌斯有界的一些必要条件。结果表明,狄利克雷位移不是莫比乌斯有界的。
更新日期:2020-05-20
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