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

Abstract

Let Möb denote the group of biholomorphic automorphisms of the unit disc and $(\mbox{Möb} \cdot T)$ be the orbit of a Hilbert space operator $T$ under the action of Möb. If the quotient $(\mbox{Möb} \cdot T)/\sim$, where $\sim$ 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.

24 pages