On the image and the kernel of a generalized derivation (Q1386498)

Let \(H_1\), \(H_2\) be Hilbert spaces, \(A\in L(H_1)\), \(B\in L(H_2)\), and \[ \delta_{A,B}(x)= AX- XB,\quad\forall X\in L(H_2, H_1). \] Among other results, the authors prove that: If there exists a quadratic polynomial \(p\) such that \(p(A)\) and \(p(B)\) are normal, then \(\overline{R(\delta_{A,B})}\cap \text{Ker}(\delta_{A^*,B^*})= \{0\}\); where \(\overline{R(\delta_{A,B})}\) is the closure of the image \(R(\delta_{A,B})\) of \(\delta_{A,B}\). When \(A= B\), we get a result of Yang Ho.