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range and $C \\in L(\\mathcal{H}),$ we study the following weighted approximation problem: analize the existence of \\begin{equation}\\label{eqa1} \\underset{X \\in L(\\mathcal{H})}{min}\\Vert AXB-C \\Vert_{p,W}, \\ \\ \\ \\ (1) \\end{equation} where $\\Vert X \\Vert_{p,W}=\\Vert W^{1/2}X \\Vert_{p}.$ We also study the related operator 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