The Stokes operator is defined by $Au=-P_H\Delta u$, where $u\in D(A)=H^2(\Omega)\cap V$ and $P_H$ is the Leray's projector. From Spectral Theorem, we can obtain an orthonormal basis of eigenvectors $(w_i)_{i\in \mathbb{N}}$ of H, i.e., $$Aw_j=\lambda_iw_i,\ \forall i\in\mathbb{N}.$$
I was reading a paper where the authors uses the following property
$$-\Delta w_i=\lambda_i w_i,\tag{1}\label{1}$$
i.e, $w_i$ is an eigenvector of the Laplacian too. Someone told me that's because $P_H$ and $-\Delta$ commutes for the eigenvectors.
First I would like to know if they commutes for the eigenvectors of the Stokes operator and why. If they don't, I would like to know if \eqref{1} is true. I will be satisfied with a good reference, I was not able to find one.
I've read they do not commute, in general. In the absence of boundaries, for example, they commute, but I couldn't find anything with respect to the eigenvectors.