Quasinilpotent linear relations: Insights, structure and applications

This paper explores the concepts of nilpotent and quasinilpotent linear relations in the context of Banach spaces. We establish sufficient conditions under which a quasinilpotent linear relation can be classified as nilpotent, focusing on the finiteness of ascent and descent and the closure of iterated images. We then introduce and analyze the class of meromorphic linear relations, emphasizing conditions that lead to their classification as quasinilpotent. Furthermore, we expand on the notion of quasinilpotent linear relations by introducing the concept of regularity and defining R-quasinilpotent relations. As an application, we investigate generalized Drazin invertible relations related to a given regularity.