B-Weyl spectrum and poles of the resolvent
A bounded linear operator \(T\) on a Banach space \(X\) is called a \(B\)-Fredholm operator if for some integer \(n\), the range space \(R(T^n)\) is closed and \(T: R(T^n) \rightarrow R(T^n)\) is a Fredholm operator. When the index is \(0\), \(T\) is called a \(B\)-Weyl operator. In this case, the \(B\)-Weyl spectrum is defined by \(\sigma_{BW}(T) = \{ \lambda : T - \lambda I\) is not a \(B\)-Weyl operator\}. Let \(E(T)\) denote the set of isolated eigenvalues of \(\sigma(T)\). In [Proc. Am. Math. Soc. 130, 1717--1723 (2002; Zbl 0996.47015)], the author has shown that \(\sigma_{BW}(T) = \sigma(T)-E(T)\) for a normal operator \(T\) on a Hilbert space. In the present paper, the author shows that in the context of Banach spaces, \(\sigma_{BW} \subset \sigma(T) - E(T)\) if and only if \(E(T)\) coincides with the set of all the poles of the resolvent of \(T\).
scientific article; zbMATH DE number 1846009