Another note on Weyl’s theorem

Abstract

“Weyl’s theorem holds" for an operator T T on a Banach space X X when the complement in the spectrum of the “Weyl spectrum" coincides with the isolated points of spectrum which are eigenvalues of finite multiplicity. This is close to, but not quite the same as, equality between the Weyl spectrum and the “Browder spectrum", which in turn ought to, but does not, guarantee the spectral mapping theorem for the Weyl spectrum of polynomials in T T . In this note we try to explore these distinctions.