Normality conditions for the matrix operator $X \rightarrow AX + X^{*}B$

Abstract

The operator defined by the left-hand side of the matrix equation

$$\begin{aligned} AX + X^{*} B = C \end{aligned}$$

is not linear in the space of complex matrices $X$. However, it becomes a linear operator if every matrix $X$ is replaced by the pair of real matrices of the same size, namely, by its real and imaginary parts. Introducing an appropriate scalar product in the resulting real space, we find necessary and sufficient conditions for this operator to be normal.