  Given a Banach space operator $A$, the isolated eigenvalues $E(A)$ and the
poles $\Pi(A)$ (resp., eigenvalues $E^a(A)$ which are isolated points of the
approximate point spectrum and the left ploles $\Pi^a(A)$) of the spectrum of
$A$ satisfy $\Pi(A)\subseteq E(A)$ (resp., $\Pi^a(A)\subseteq E^a(A)$), and the
reverse inclusion holds if and only if $E(A)$ (resp., $E^a(A)$) has empty
intersection with the B-Weyl spectrum (resp., upper B-Weyl spectrum) of $A$.
Evidently $\Pi(A)\subseteq E^a(A)$, but no such inclusion exists for $E(A)$ and
$\Pi^a(A)$. The study of identities $E(A)=\Pi^a(A)$ and $E^a(A)=\Pi(A)$, and
their stability under perturbation by commuting Riesz operators, has been of
some interest in the recent past. This paper studies the stability of these
identities under perturbation by (non-commuting) compact operators. Examples of
analytic Toeplitz operators and operators satisfying the abstract shift
condition are considered.
