Conjugating by singular operators: On the boundedness of similarity transforms near singular points

We consider the question of, given operators $A$, $Z$ and a sequence of invertible operators $U_n\to Z$, whether the sequence $U_nAU_n^{-1}$ is bounded in norm, as well as generalizations of this where $U_nAU_n^{-1}$ is modified by some bounded linear map on bounded linear operators. In the setting of Hilbert spaces, we provide a complete classification in terms of algebraic criteria of those $A$ for which such a sequence exists, as long as $Z$ is of generalized index zero, which always holds in finite-dimensional contexts. In the process, we prove that particular coefficients arising in inverses of certain good paths going to $Z$ can also be classified in terms of an entirely algebraic criterion.