The polar decomposition of the product of three operators

TL;DR

This paper investigates the polar decomposition of the product of three operators on Hilbert C*-modules, establishing relationships and formulas for perturbations and characterizations.

Contribution

It introduces new formulas and characterizations for the polar decomposition of the product of three operators in the context of Hilbert C*-modules.

Findings

Derived a formula for polar decomposition of a perturbed operator

Clarified relationships between polar decompositions of three operators

Provided characterizations of polar decompositions in this setting

Abstract

In the setting of adjointable operators on Hilbert $C^*$-modules, this paper deals with the polar decomposition of the product of three operators. The relationship between the polar decompositions associated with three operators is clarified. Based on this relationship, a formula for the polar decomposition of a multiplicative perturbation of an operator is provided. In addition, some characterizations of the polar decomposition associated with three operators are provided.