The property $(\omega \pi )$ as a generalization of the a-Weyl theorem

In this paper, for a bounded linear operator defined on a complex Banach space of infinite dimension, we consider the set of isolated points in its approximate point spectrum, which are eigenvalues of finite multiplicity; this set can be equal to the spectrum of the operator but without its upper semi-Fredholm spectrum, and this relation or equality defines in the literature a new spectral property called the property $(\omega \pi )$ and is a generalization of the classical a-Weyl theorem. We establish some characterizations and consequences about the property $(\omega \pi )$, some with topological aspects. Furthermore, we study this property through the Riesz functional calculus. Part of the spectral structure of a linear operator verifying property $(\omega \pi )$ is described, obtaining some associated properties.