# Weighted weak group inverse for Hilbert space operators

## TL;DR

This paper introduces the weighted weak group inverse for operators on Hilbert spaces, extending the concept from matrices, with new characterizations, representations, and applications to binary relations.

## Contribution

It presents the first definition and analysis of the weighted weak group inverse for Hilbert space operators, expanding the theory beyond matrices.

## Key findings

- Defined the weighted weak group inverse for Hilbert space operators
- Provided characterizations and representations of the inverse
- Applied the inverse to define and analyze binary relations

## Abstract

We present the weighted weak group inverse, which is a new generalized inverse of operators between two Hilbert spaces, introduced to extend weak group inverse for square matrices. Some characterizations and representations of the weighted weak group inverse are investigated. We also apply these results to define and study the weak group inverse for a Hilbert space operator. Using the weak group inverse, we define and characterize various binary relations.