  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.
