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Let M\"{o}b denote the group of biholomorphic automorphisms of the unit discand $(\mbox{M\"{o}b} \cdot T)$ 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 theoperator $T$ is said to be weakly homogeneous. In this paper, we obtain acriterion to determine if the operator $M_z$ of multiplication by thecoordinate function $z$ on a reproducing kernel Hilbert space is weaklyhomogeneous. We use this to show that there exists a M\"{o}bius bounded weaklyhomogeneous operator which is not similar to any homogeneous operator,answering a question of Bagchi and Misra in the negative. Some necessaryconditions for the M\"{o}bius boundedness of a weighted shift are alsoobtained. As a consequence, it is shown that the Dirichlet shift is notM\"{o}bius bounded.

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