Paranormal operators on Banach spaces

Abstract

In this note we show that a paranormal operator T on a Banach space satisfies Weyl's theorem. This is accomplished by showing that(i) every isolated point of its spectrum is an eigenvalue and the corresponding eigenspace has invariant complement,(ii) for α ≠ 0, Ker(T-α) ⊥ Ker (T-β) (in the sense of Birkhoff) whenever β ≠ α.