Criteria for the property (UWE) and the a-Weyl theorem

In this paper, the property (UWE) and the a-Weyl theorem for bounded linear operators are studied in terms of the property of topological uniform descent. Sufficient and necessary conditions for a bounded linear operator defined on a Hilbert space to have the property (UWE) and satisfy the a-Weyl theorem are established. In addition, new criteria for the fulfillment of the property (UWE) and the a-Weyl theorem for an operator function are discussed. As a consequence of the main theorem, results on the stability of the property (UWE) and the a-Weyl theorem are obtained.