Finite sections: stability, spectral pollution and asymptotics of condition numbers and pseudospectra

Abstract

The stability of an approximating sequence $(A_n)$ for an operator $A$ usually requires, besides invertibility of $A$, the invertibility of further operators, say $B,C,\dots$, that are well-associated to the sequence $(A_n)$. We study this set, $\{A,B,C,\dots \}$, of so-called stability indicators of $(A_n)$ and connect it to the asymptotics of $\Vert A_n\Vert$, $\Vert A_n^{-1}\Vert$ and $\kappa (A_n)=\Vert A_n\Vert \Vert A_n^{-1}\Vert$ as well as to spectral pollution by showing that $\mathrm{limsup}{\mathrm{Spec}}_{\epsilon }{A}_n={\mathrm{Spec}}_{\epsilon }A\cup {\mathrm{Spec}}_{\epsilon }B\cup {\mathrm{Spec}}_{\epsilon }C\cup \dots$. We further specify, for each of $\Vert A_n\Vert$, $\Vert A_n^{-1}\Vert$, $\kappa (A_n)$ and ${\mathrm{Spec}}_{\epsilon }{A}_n$, under which conditions even convergence applies.