# When do finite dimensional approximations approximate the spectral absicssa of a linear operator?

I apologize if the following is trivial for experts in the field. If so, please feel free to refer me instead to any proper references.

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.

Are there any results which address the convergence of the pseudospectral abscissas of finite dimensional approximations 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 $\mathcal{L}$ (in my case, e.g., finite difference or spectral discretizations), 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, what can we deduce about the convergence $\sigma_\varepsilon(M_n) \rightarrow \sigma_\varepsilon(\mathcal{L})$? 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)?

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

EDIT: I have found the article Discrete Approximation of Unbounded Operators and Approximation of their Spectra by Wolff. It seems to address the approximation of the approximate point spectrum by finite dimensional approximations.