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 $\|A_n\|$, $\|A_n^{-1}\|$ and $\kappa(A_n)=\|A_n\|\|A_n^{-1}\|$ as well as to spectral pollution by showing that $\limsup {\rm Spec}_\varepsilon A_n= {\rm Spec}_\varepsilon A\cup{\rm Spec}_\varepsilon B\cup{\rm Spec}_\varepsilon C\cup\dots$. We further specify, for each of $\|A_n\|$, $\|A_n^{-1}\|$, $\kappa(A_n)$ and ${\rm Spec}_\varepsilon A_n$, under which conditions even convergence applies.

翻译：操作符$A$的逼近序列$(A_n)$的稳定性通常除了要求$A$可逆外，还需与序列$(A_n)$紧密关联的其他操作符（如$B, C, \dots$）可逆。本文研究$(A_n)$的这类所谓稳定性指示集$\{A,B,C,\dots\}$，通过证明$\limsup {\rm Spec}_\varepsilon A_n= {\rm Spec}_\varepsilon A\cup{\rm Spec}_\varepsilon B\cup{\rm Spec}_\varepsilon C\cup\dots$，将其与$\|A_n\|$、$\|A_n^{-1}\|$及$\kappa(A_n)=\|A_n\|\|A_n^{-1}\|$的渐近性以及谱污染问题建立关联。我们进一步针对$\|A_n\|$、$\|A_n^{-1}\|$、$\kappa(A_n)$和${\rm Spec}_\varepsilon A_n$，明确了各自满足收敛性的具体条件。