Right and left quotient of two bounded operators on Hilbert spaces

TL;DR

This paper introduces and characterizes left and right quotients of bounded operators on Hilbert spaces using Moore-Penrose inverses, establishing their properties and explicit computation formulas.

Contribution

It defines and parametrizes left and right quotients of bounded operators on Hilbert spaces, linking them via Moore-Penrose inverses and providing explicit formulas for their computation.

Findings

The adjoint of a left quotient is a right quotient and vice versa.

Explicit formulas for quotients under adjoint, sum, and product operations.

Parametrization of quotients using Moore-Penrose inverse.

Abstract

We define a left quotient as well as a right quotient of two bounded operators between Hilbert spaces, and we parametrize these two concepts using the Moore-Penrose inverse. In particular, we show that the adjoint of a left quotient is a right quotient and conversely. An explicit formulae for computing left (resp. right) quotient which correspond to adjoint, sum, and product of given left (resp. right) quotient of two bounded operators are also shown.