Let $M$ be a compact Riemanniam manifold and $\Delta$ the Laplace-Beltrami operator. Let $\lambda_0 \leq \ldots \leq \lambda_j \leq \ldots$ be the eigenvalues and $\phi_0, \ldots, \phi_j, \ldots$ be the eigenfunctions of the Laplace-Beltrami operator. We know that we can construct a basis for the space $L^2(M)$ by using the eigenfunctions of the Laplacian-Beltrami operator.
Now I can ask my question. Are there differential operators $P : C^\infty(M) \to C^\infty(M)$ on $M$ that can be written as the following expression?
$$ Pf = \sum_{j = 0} g(\lambda_j) \langle f, \phi_j \rangle \phi_j.$$
I know that if $g$ is a polynomial or an analytic function, then $g(\Delta)$ can be written as above. But I'm looking for other interesting differential operators. For example, are there any vector field that can be written as above?
For me it would be interesting to find examples of such operators compact Manifold.