Is there such subsequence in $L^2(\Bbb{R}^2)$.

Question

We note by $||.||_T=||.||_2+||T.||$ the graph norm of an unbounded operator $T$ with domain $D(T)=\{u\in L^2(\Bbb{R}^2), Tu\in L^2(\Bbb{R}^2)\}$ and $||u||^2_2=\int_{\Bbb{R}^2}u(x,y)\overline{u(x,y)} dxdy$.

Put $L=-\Delta_{\Bbb{R}^2}-(x^2+y^2)$ and $R=x\partial_y -y\partial_x$.

Let $(f_n)$ be a sequence in $D(R)\cap D(L)$ such that $||(f_n)||_R<C$ and $||(f_n)||_L<C$ for some constant $C>0$.

Can I find a subsequence $(f_{n_k})$ of $(f_n)$ such that $(f_{n_k})$ converge in $L^2(\Bbb{R}^2)$.

Thanks.