Geometric and semigeometric approximation of spectral projections (Q1820379)

Let T be a closed operator in a Banach space. Assume that the contour \(\Gamma\) isolates a simple eigenvalue \(\Lambda\) from the rest of the spectrum of T, and let P be the corresponding spectral projector. Consider a sequence of closed operators \(T_ n\) such that (i) \(\| T_ n-T\| \to 0\). Let \(P_ n\) be the spectral projector of \(T_ n\) belonging to the same domain; ultimately rank \(P_ n=1\), too. The authors discuss approximating \(PP_ n-P_ n\) by truncating it Neumann series in powers of \(T-T_ n\) at the term \((T-T_ n)^ j\). (Intuition leads one to ask about \(P-P_ n\), but \(PP_ n-P_ n\) is simpler and yields just as much.) Let the approximant to \(PP_ be\) called \(P^ j_ n\). It is proved that for n fixed sufficiently large, as \(j\to \infty\) we have \(\| PP_ n-P^ j_ n\| =O(\gamma^{j+1})\) for a computable \(\gamma\). A similar, weaker conclusion holds in case (i) is replaced by the hypothesis that \(T_ n\to T\) in the collectively compact sense.