# Approximating the spectrum of a non-normal, non-local differential operator

I would like to compute the spectrum of a known non-normal, non-local differential operator. This operator arises as the linearization of a nonlinear PDE around an equilibrium solution. I am concerned with stability of this equilibrium solution, so in particular I am interested in the spectral abscissa.

In order to build intuition, I intend to approximate the spectrum by finite dimensional eigenvalue problems (e.g., via finite difference approximations).

Since the operator is non-normal, I expect that these approximations are not well-behaved under perturbations. As such, I believe it is reasonable to consider the pseudospectra of these finite dimensional problems.

Are there any results which address the convergence of the pseudospectral abscissas to the spectral abscissa of the original (infinite dimensional) operator? That is, letting $\mathcal{L}$ be an unbounded operator on a Banach space and $M_n$ be finite dimensional approximations of $\mathcal{L}$, with $\Lambda$ and $\Lambda_\varepsilon$ the spectral and pseudospectral abscissas, when is it true that

$$\lim_{\varepsilon\to 0} \lim_{n \to \infty} \Lambda_\varepsilon(M_n) = \Lambda(\mathcal{L})?$$

More generally, when is it true that $\sigma_\varepsilon(M_n) \rightarrow \sigma_\varepsilon(\mathcal{L}) \rightarrow \sigma(\mathcal{L})$? Since it is known in general that $\sigma(M_n) \not\rightarrow \sigma(\mathcal{L})$, is it possible to bypass this limit via pseudospectra (see diagram below)?

$$\begin{array}{ccc} \sigma_\varepsilon(M_n) & \leftarrow & \sigma(M_n) \\ \downarrow & & \downarrow \\ \sigma_\varepsilon(\mathcal{L}) & \rightarrow & \sigma(\mathcal{L}) \\ \end{array}$$