A note on commutativity up to a factor of bounded operators

In this note, we explore commutativity up to a factor $AB=\lambda BA$ for bounded operators $A$ and $B$ in a complex Hilbert space. Conditions on possible values of the factor $\lambda$ are formulated and shown to depend on spectral properties of the operators. Commutativity up to a unitary factor is considered. In some cases, we obtain some properties of the solution space of the operator equation $AX=\lambda XA$ and explore the structures of $A$ and $B$ that satisfy $AB=\lambda BA$ for some $\lambda \in \mathbb{C}\setminus \{ 0 \}.$ A quantum effect is an operator $A$on a complex Hilbert space that satisfies $0\leq A \leq I.$ The sequential product of quantum effects $A$ and $B$ is defined by $A\circ B=A^{\frac{1}{2}}BA^{\frac{1}{2}}.$ We also obtain properties of the sequential product.