GD1 inverse and 1GD inverse for bounded operators on Banach spaces

Mosić and Djordjević introduced the notation of the gDMP inverse for a linear operator on Hilbert space in Mosić and Djordjević (J Spectr Theory 8(2):555–573, 2018) by considering generalized Drazin inverse with the Moore-Penrose inverse. This article introduces two new classes of inverses: GD1 (generalized Drazin and inner) inverse and 1GD (inner and generalized Drazin) inverse for Banach space operators. The existence and uniqueness of the GD1 (also 1GD) inverse are discussed along with some properties through core-quasinilpotent decomposition and closed range decomposition operator. We further establish a few explicit representations of the GD1 inverse and their interconnections with generalized Drazin inverse. In addition, we discuss a few properties of GD1 (also 1GD) inverse through binary relation.