Moore-Penrose inverse and partial orders on Hilbert space operators

TL;DR

This paper investigates the properties of the Moore-Penrose inverse of bounded linear operators on Hilbert spaces, focusing on monotonicity, reverse order law, and weighted inverse interpretations.

Contribution

It provides new insights into the monotonicity of the Moore-Penrose inverse under various operator orders and examines conditions for the reverse order law to hold.

Findings

Monotonicity properties of the Moore-Penrose inverse under multiple orders.

Conditions under which the reverse order law $B^ager A^ager = (AB)^ager$ is valid.

Interpretation of $B^ager A^ager$ as weighted inverses of $AB$.

Abstract

In this article we explore several aspects concerning to the Moore-Penrose inverse of a bounded linear operator. On the one hand, we study monotonicity properties of the Moore-Penrose inverse with respect to the L\"owner, star, minus, sharp and diamond orders. On the other hand, we analyze the validity of the reverse order law, $B^\dagger A^\dagger=(AB)^\dagger$, under hypothesis of operator ranges and also under hypothesis of order operators. Finally, we study the operator $B^\dagger A^\dagger$ as different weighted inverses of $AB$.