Let $\Omega$ be a bounded open subset of $\mathbb{R}^3$ and $n$ the outward-pointing unit normal to the boundary of $\Omega$. We know that the space $(L^2(\Omega))^3$ of square integrable functions admits the following orthonormal decomposition: $$ (L^2(\Omega))^3=\nabla H_0^1(\Omega) \oplus H(div\,0,\Omega), $$ where
$div$ is the divergence operator, $\nabla$ the gradient operator,
$H^1_0(\Omega)$ is the usual Sobolev space,
and $H(div\,0,\Omega):=\{u\in (L^2(\Omega))^3|div(u)=0\}$.
We want to construct a bounded operator defined on $(L^2(\Omega))^3$ such that its restriction on the subspace $\nabla H_0^1(\Omega)$ is the identity operator and the other one on the subspace $H(div\,0,\Omega)$ is the null operator. Thanks.