Almost stable spectra and limits of similarities

Abstract

Suppose that {D n } is a sequence of invertible operators on a Hilbert space, andD n T D −1 converges in norm toT 0. Recently, H. Bercovici, C. Foias, and A. Tannenbaum have shown that if {D ±1 ∶n=1, 2,...} is contained in a finite dimensional subspace of operators, thenT andT 0 must have the same spectral radius. Using this result, R. Teodorescu proved that the resolvents ofT andT 0 have the same unbounded component. We show that in fact the spectra differ only by certain eigenvalues ofT 0, and the spectrum ofT 0 is obtained by “filling in holes” in the spectrum ofT; i.e., by adjoining (all, some, or none of the) bounded components of the resolvent ofT to the spectrum ofT.