Symmetric nonnegative definite solutions of certain linear matrix equations on a subspace

For a given matrix $G\in {\mathbb{R}}^{l\times n}$, define a subspace $\mathrm{\Omega }$ by $\mathrm{\Omega }=\left\{z\in {\mathbb{R}}^n\mid Gz=0\right\}$, and let $\mathcal{H}=\left\{K\in {\mathbb{R}}^{n\times n}\mid (x,Ky)=(Kx,y),x^{\mathrm{\top }}Kx\ge 0,\mathrm{\forall }x,y\in \mathrm{\Omega }\right\}$. Thus, $\mathcal{H}$ consists of all symmetric nonnegative definite matrices on the subspace $\mathrm{\Omega }$. In this paper, we investigate the existence of $X\in \mathcal{H}$ satisfying the linear matrix equations $AX=C,XB=D$, and $AXB=D$. A necessary and sufficient condition for solvability and an explicit representation of the general solution are obtained. Finally, two numerical examples are provided to demonstrate the accuracy of the results.