Polar decomposition and characterization of binormal operators

Abstract

We illustrate the matrix representation of the closed range operator that enables us to determine
the polar decomposition with respect to the orthogonal complemented submodules. This result
proves that the reverse order law for the Moore–Penrose inverse of operators holds. Also, it is given
some new characterizations of the binormal operators via the generalized Aluthge transformation. New
characterizations of the binormal operators enable us to obtain equivalent conditions when the inner
product of the binormal operator with its generalized Aluthge transformation is positive in the general
setting of adjointable operators on Hilbert C∗-modules.