Isolated eigenvalues, poles and compact perturbations

Analyse Fonctionnelle

Salle Kampé de Fériet M2
B. P. Duggal
Vendredi, 14 Décembre, 2018 - 14:00 - 15:00
The isolated eigenvalues $E(A)$, the poles $\Pi(A)$, eigenvalues $E^a(A)$ (isolated points of the approximate point spectrum $\sigma_a(A)$ which are eigenvalues) and the left poles $\Pi^a(A)$ of the spectrum of a Banach space operator $A$ satisfy $\Pi(A)\subseteq E(A)\subseteq E^a(A)$ and $\Pi(A)\subseteq\Pi^a(A)\subseteq E^a(A)$). The reverse inclusions, in particular the properties $$ (P1): E(A)=\Pi^a(A) ; \hspace{4mm} (P2): E^a(A)=\Pi(A),$$ and their stability under commuting Riesz perturbations have been a topic of some interest in the recent past. If we let $\sigma_w(A)$, $\sigma_{Bw}(A)$, $\sigma_{aw}(A)$ and $\sigma_{uBw}(A)$ denote,respectively, the Weyl, B-Weyl, upper Weyl and upper B-Weyl spectrum of $A$, then $A\in (P1)\Longleftrightarrow E(A)\cap\sigma_{uBw}(A)=\emptyset$ and $A\in (P2)\Longleftrightarrow E^a(A)\cap\sigma_{uBw}(A)=\emptyset$; if $R$ is Riesz, $AR-RA=0$, ${\textrm{iso}}\sigma_a(A)={\textrm{iso}}\sigma_a(A+R)$ and ${\textrm{iso}}\sigma_{aw}(A)=\emptyset$, then $A\in (P_i)\Longleftrightarrow A+R\in (P_i)$, $i=1,2$. Neither of the properties $P(1)$ and $P(2)$, or their finite dimensional versions $(P1)':E_0(A)=\Pi^a_0(A)$ and $(P2)':E^a_0(A)=\Pi_0(A) $, travels well under perturbation by (not necessarily commuting) compact operators $K$. Typically, one has: Suppose ${\textrm{iso}}\sigma_a(A+K)={\textrm{iso}}\sigma_a(A)$. If ${\textrm{iso}}\sigma_w(A)\cap\{\sigma(A)\setminus\sigma_{Bw}(A)\}=\emptyset$, then $A\in (P1)\Longleftrightarrow A\in (P1)'$ and $A\in (P1)\Longleftrightarrow A+K\in (P1)'$ iff $E_0(A+K)\subseteq E_0(A)$; if $\sigma_{Bw}(A)\setminus\sigma_{uBw}(A)=\sigma_{Bw}(A+K)\setminus\sigma_{uBw}(A+K)$ and ${\textrm{iso}}\sigma_a(A)\cap\{\sigma_{Bw}(A+K)\setminus\sigma_{Bw}(A)\}=\emptyset$, then a sufficient condition for $A\in (Pi)\Longrightarrow A+K\in (Pi)$, $i=1,2$, is that ${\textrm{iso}}\sigma_a(A)\cap\sigma_{uBw}(A)=\emptyset$.