Abstract
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 Mz 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.
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Acknowledgement
The author would like to thank G. Misra for many fruitful discussions and suggestions in the preparation of this paper.
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This work was supported by CSIR SPM Fellowship (Ref. No. SPM-07/079(0242)/2016-EMR-I). The results in this paper are from the PhD thesis of the author submitted to the Indian Institute of Science.
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Ghara, S. The orbit of a bounded operator under the Möbius group modulo similarity equivalence. Isr. J. Math. 238, 167–207 (2020). https://doi.org/10.1007/s11856-020-2016-x
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DOI: https://doi.org/10.1007/s11856-020-2016-x