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

Abstract

Let Mob denote the group of biholomorphic automorphisms of the unit disc and (Mob · T) be the orbit of a Hilbert space operator T under the action of Mob. 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 Mobius 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 Mobius boundedness of a weighted shift are also obtained. As a consequence, it is shown that the Dirichlet shift is not Mobius bounded.