Let $g\in H_0^1(\Omega)\cap W^{2,\infty}(\Omega),$ and let us define the operator $B : y \to g y$ from $H:=H_0^1(\Omega)\cap H^2(\Omega)$ to $H$, which we endowed with norm $|u|=(\|u\|^2 +\|\Delta u\|^2)^{\frac{1}{2}}$, where $\|\cdot\|$ is the norm of $L^2(\Omega).$
I want to explicit the adjoint operator $B^*$ of $B$ in $(H,|.|)$.