Developing Reverse Order Law for the Moore–Penrose Inverse with the Product of Three Linear Operators

Yang Qi,Yu Yaoming,Liu Xiaoji,Valerii Obukhovskii

doi:10.1155/2021/6585951

Abstract

In this paper, we study the reverse order law for the Moore–Penrose inverse of the product of three bounded linear operators in Hilbert spaces. We first present some equivalent conditions for the existence of the reverse order law A B C † = C † B † A † . Moreover, several equivalent statements of ℛ A A ∗ A B C = ℛ A B C and ℛ C ∗ C A B C ∗ = ℛ A B C ∗ are also deducted by the theory of operators.

Highlights

Let L(X, Y) denote the set of all bounded linear operators from Hilbert space X to Hilbert space Y
E reverse order law for the generalized inverse of a product of matrices or bounded linear operators yields a class of interesting fundamental problems in the theory of the generalized inverse of matrices or operators
Panigrahy and Mishra presented some properties for the weighted Moore–Penrose inverse for an arbitrary order tensor via the Einstein product [12]; they proposed the expression for the Moore–Penrose inverse of the product of two tensors via the Einstein product [13, 14]

Summary

Introduction

Let L(X, Y) denote the set of all bounded linear operators from Hilbert space X to Hilbert space Y. E reverse order law for the generalized inverse of a product of matrices or bounded linear operators yields a class of interesting fundamental problems in the theory of the generalized inverse of matrices or operators. It has been widely studied since the middle 1960s. Is motivates us to have a further study concerning the reverse order law of three operators in Hilbert space and to extend some results in [4, 8]. ABC will be well-defined in what follows

Lemmas

Reverse Order Laws