Let $M$ be a $3\times 3$ real invertible diagonal matrix and $H$ a Hilbert space of infinite dimension (for example, we can take $H$ as the space of square integrable functions over a bounded lipschitz domain $\Omega\subset\mathbb{R}^3$).  

I want to determine the essential spectrum of $M$ as a multiplication operator acting from $H^3$ to itself.  

Does it consist of the eigenvalues of $M$? Or its numerical range ?
Thank you.



Essential spectrum of constant invertible diagonal matrix acting on a product of Hilbert spaces
$H^3$ is $\mathbb{C}^3 \otimes H$ and the action of $M$ on it is just the usual action on the first tensor factor. It is a completely straighforward exercise to prove that the $\lambda$-eigenspace of $M\otimes\operatorname{id}\_H$ is exactly $\operatorname{Eig}\_\lambda(M) \otimes H$ and similarly for generalized eigenspaces. In particular: The spectrum of $M\otimes\operatorname{id}\_H$ is exactly the same as the spectrum of $M$, i.e. discrete consisting precisely of the eigenvalues.


