Summary: Let \(H_ 1\) and \(H_ 2\) be separable Hilbert spaces, \(B(H_ 2,H_ 1)\) the set of all bounded linear transformations from \(H_ 2\) to \(H_ 1\). Let A and B belong to \(B(H_ 1)\) and \(B(H_ 2)\) respectively. Define an operator \(\delta_{BA}\) on \(B(H_ 2,H_ 1)\) by \(\delta_{AB}(x)=AX- XB\), \(X\in B(H_ 2,H_ 1)\), we call it a generalized derivation. \(R(\delta_{AB})\) denotes the range of \(\delta_{AB}\). There is a basic and long-time suspended problem about the generalized derivation ranges, that is: When is the range \(R(\delta_{AB})\) norm closed? In 1976, it is C. Apostol, who excellently characterized the case when \(A=B\). In 1980, L. A. Fialkow obtained some results on this problem in some special cases such as: A and B are hyponormal or compact operators [J. Oper. Theory 3, 89-113 (1980; Zbl 0443.47004)]. He also made a few remarks on A and B being nilpotent operators (especially with order 2). In what follows, we give the characterization of this problem when it is defined by a dominant operator and a codominant operator (i.e. the adjoint of a dominant one). In addition, we construct an example which settles Fialkow’s two open questions negatively.