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We give an answer to the following problem: Given two linear operators A and B such that BA and A verify some property P, then when does B verify the same property P? Of course, we have to assume that B satisfies some condition Q independent of (or weaker than) P. This problem is solved in the setting of both bounded and unbounded operators on a Hilbert space. Some interesting counterexamples are also given

An implicit division of bounded and unbounded linear operators which preserves their properties