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

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 $\mathrm{\parallel }{A}_n\mathrm{\parallel }$, $\mathrm{\parallel }{A}_n^{-1}\mathrm{\parallel }$ and $\kappa (A_n)=\mathrm{\parallel }{A}_n\mathrm{\parallel }\mathrm{\parallel }{A}_n^{-1}\mathrm{\parallel }$ as well as to spectral pollution by showing that $\mathrm{lim\hspace{0.17em}sup}{\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}}_{\epsilon }{A}_n={\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}}_{\epsilon }A\cup {\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}}_{\epsilon }B\cup {\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}}_{\epsilon }C\cup \dots$. We further specify, for each of $\mathrm{\parallel }{A}_n\mathrm{\parallel }$, $\mathrm{\parallel }{A}_n^{-1}\mathrm{\parallel }$, $\kappa (A_n)$ and ${\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}}_{\epsilon }{A}_n$, under which conditions even convergence applies.