Commutativity up to a factor for bounded and unbounded operators (Q2252456)

Let \(A,B\) be linear operators on a complex Hilbert space. We say that \(A,B\) commute up to a factor if \(AB=\lambda BA\) holds for some scalar \(\lambda\in {\mathbb C}\). The authors further investigate the problem of commutativity up to a factor in the setting of bounded and unbounded linear operators in a complex Hilbert space. The following results are proved. {\parindent=0.6cm \begin{itemize}\item[{\(\bullet\)}] Let \(A\) and \(B\) be two bounded normal operators such that \(AB=\lambda BA\neq 0\), \(\lambda\in {\mathbb C}\). Then \(AB\) and \(BA\) are normal for any nonzero \(\lambda\). \item [{\(\bullet\)}] Let \(A\) and \(B\) be two normal operators where \(B\) is bounded and \(BA\subset \lambda AB\neq 0\), \(\lambda\in {\mathbb C}\). Then \(AB\) is normal iff \(|\lambda|=1\). \item [{\(\bullet\)}] Let \(A, N, M\) are unbounded invertible operators where \(N\) and \(M\) are normal. If \(AN=MA\), then \(A^*M=NA^*\) and \(AN^*=M^*A\). \end{itemize}} Applications to the case of self-adjoint operators are also included. The paper contains a conjecture on the commutativity of self-adjoint operators.