GD1 inverse and 1GD inverse for Hilbert space operators

TL;DR

This paper introduces two new classes of inverses for Hilbert space operators, the GD1 and 1GD inverses, exploring their properties, existence, uniqueness, and relationships with existing inverses.

Contribution

The paper defines the GD1 and 1GD inverses, establishes their properties, and connects them with the generalized Drazin inverse, expanding the theory of operator inverses.

Findings

Existence and uniqueness of GD1 and 1GD inverses are established.

Explicit representations and properties of these inverses are provided.

Connections with the generalized Drazin inverse are demonstrated.

Abstract

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.