Riesz points and Weyl's theorem
A Riesz point for a bounded linear operator \(T\) on a complex Banach space is simply an isolated point of the spectrum \(\sigma(T)\) such that the corresponding spectral projection has finite rank. The Weyl spectrum \(\sigma_W(T)\) is that part of \(\sigma(T)\) that remains unchanged under every compact perturbation of \(T\). In the paper under review, the following two conditions on \(T\) are considered: I. \(\sigma_W(T)\) equals the complement in \(\sigma(T)\) of the set of isolated eigenvalues having finite-dimensional eigenspace (``Weyl's theorem''). II. \(\sigma_W(T)\) equals the complement in \(\sigma(T)\) of the set of Riesz points of \(T\) (``Version II of Weyl's theorem''). One first establishes some connections between the these two conditions on \(T\). Several closure properties are then pointed out for the class of operators enjoying II. After that, one proves that, if \(\sigma_W(T)\) is simply connected, then Version II of Weyl's theorem holds for every compact perturbation of \(T\). The converse assertion is also considered. At the final of paper, an example is presented in order to ``draw some conclusions which contrast Weyl's theorem and Version II'' (in the author's words).
scientific article; zbMATH DE number 1333092
scientific article