  Mosic and Djordjevic introduced the notation of the gDMP inverse for Hilbert
space operators in [J. Spectr. Theory, 8(2):555-573, 2018] by considering
generalized Drazin inverse with the Moore-Penrose inverse. This paper
introduces two new classes of inverses: GD1 (generalized Drazin and inner)
inverse and 1GD (inner and generalized Drazin) inverse for Hilbert 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.

Functional Analysis;Operator Algebras