Weyl's theorem through finite ascent property (Q1885515)

For a Banach space operator \(T\) let \(\sigma(T)\) denote its spectrum, \(\sigma_w(T)\) its Weyl spectrum, let \(\pi_{00}(T)\) stand for the set of eigenvalues of \(T\) of finite geometric multiplicity and set \(\pi_{00}^\#(T)=\{\lambda\in\pi_{00}(T)\colon\, (T-\lambda) \;\text{has both finite ascent and descent}\}\). The operator \(T\) satisfies Weyl's theorem if \(\sigma(T)\setminus\sigma_w(T)= \pi_{00}(T)\). The main result of the present paper is the following Theorem. An operator \(T\in B(X)\) satisfies Weyl's theorem if and only if it has the single valued extension property at all points \(\lambda\in{\mathbb C}\setminus\sigma_w(T)\) and \(\pi_{00}(T)= \pi_{00}^\#(T)\). A version of this theorem for essential spectra is also proven. The well-known sufficient condition for a Banach space operator \(T\) to satisfy Weyl's theorem is that the quasi-nilpotent part \(H_0(T-\lambda)\) of \(T-\lambda\) is equal to \((T-\lambda)^{-p}(0)\) for all complex \(\lambda\) and some positive integer \(p\geq 1\). The authors give examples of operators that satisfy the hypotheses of the Theorem but do not satisfy this condition.