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 inner product : $<u,v>_H =<u,v>_{L^2(\Omega)} + <\Delta u,\Delta v>_{L^2(\Omega)}$,
I want to explicit the adjoint operator $B^*$ of $B$ in $(H,<,>)$.