The orbit of a bounded operator under the M\"{o}bius group modulo similarity equivalence

Let M\"{o}b denote the group of biholomorphic automorphisms of the unit disc and $(\mbox{M\"{o}b} \cdot T)$ be the orbit of a Hilbert space operator $T$ under the action of M\"{o}b. If the quotient $(\mbox{M\"{o}b} \cdot T)/\sim$, 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 M\"{o}bius 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 M\"{o}bius boundedness of a weighted shift are also obtained. As a consequence, it is shown that the Dirichlet shift is not M\"{o}bius bounded.