Let $\Gamma$ be a $C^k$ $(n-1)$-dimensional hypersurface embedded in $\mathbb{R}^n$. Let $H=L^2(\Gamma)$ and $V=H^1(\Gamma)$.

Suppose that $\{v_j\}$ is a basis for $H$ and $V$ (not necessarily orthogonal).

Let $V_m = \text{span}(v_1, ..., v_m)$.

Define a projection operator $P_m:H \to V_m$ satisfying $$(P_m h - h, v_m) = 0 \qquad\text{for all $v_m \in V_m$}.$$

Is it true that $$P_m (\nabla_{\Gamma}h) = \nabla_{\Gamma}(P_mh)$$ (on the LHS, the projection operator is applied to each element of the gradient vector) and $$P_m (\Delta_{\Gamma}h) = \Delta_{\Gamma}(P_mh)?$$ Where $\nabla_{\Gamma}$ is the surface gradient (defined by $\nabla_\Gamma f= \nabla f - \nabla f \cdot \nu \nu$ with $\nu$ the unit normal) and $\Delta_\Gamma$ is the Laplace-Beltrami operator. I.e. does the projection operator and the surface gradient operator commute??