Polar decomposition and characterization of binormal operators

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.