Weyl type theorems for functions of operators (Q2902673)

Let \(X\) be a complex Banach space. For a bounded linear operator \(T\) on \(X\), let \(\sigma(T)\) be the spectrum, \(\sigma_a(T)\) be the approximate point spectrum, and \(\sigma_{ea}(T)=\cap_{K}\sigma_a(T+K)\), where the intersection is taken over all compact operators \(K\) on \(X\), the essential approximate point spectrum. Let \(\pi_{00}(T)\) be the set of isolated eigenvalues in \(\sigma(T)\) such that the kernel of \(\lambda I-T\) is finite dimensional and let \(\pi_{00}^{a}(T)\) be the set of all eigenvalues which are isolated points in \(\sigma_{a}(T)\) and such that the kernel of \(\lambda I-T\) is finite dimensional. It is said that a-Weyl's theorem holds for \(T\), briefly denoted as \(T\in (a-W)\), if \(\sigma_a(T)\setminus \sigma_{ea}(T)=\pi_{00}^{a}(T)\). If \(\sigma_a(T)\setminus \sigma_{ea}(T)=\pi_{00}(T)\), then \(T\in (\omega)\). In this paper, there are established necessary and sufficient conditions for \(T\) such that \(f(T)\in (a-W)\), respectively \(f(T)\in (\omega)\), for every function \(f\) which is holomorphic on a neighborhood of \(\sigma(T)\).