The orbit of a bounded operator under the Möbius group modulo similarity equivalence

2018/11/13 by Ghara, Soumitra · 1 citation
#30J #47B32 #47B33 #47B37 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1811.05428

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 (Möb ⋅ T)/∼, 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.

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