Some Classes Related to the Set of Generalized Drazin Invertible Linear Relations

Abstract

In this paper, we introduce and investigate several classes of linear relations on Banach spaces related to generalized Drazin invertibility. First, we focus on a subclass of the generalized Drazin invertible linear relations, namely the class of generalized strongly Drazin invertible linear relations. In particular, we derive two distinct characterizations: one in terms of a bounded projection and a quasi-nilpotent operator, and another based on a specific relationship between T and T². This, in turn, leads us to the definition and  study of the class of Hirano invertible linear relations. We then introduce the concept of weakly generalized Drazin invertibility and establish a connection between this notion and invertibility in the sense of Hirano. As an application, we present new criteria ensuring the existence of a generalized Drazin inverse for a given linear relation T.

Keywords: Linear relation; Generalized inverse; Drazin inverse; Hirano inverse.

2010 Mathematics Subject Classification. 47A06; 16U90