Generalized Inverse Preservers of Hadamard Circulant Majorization

Let $T\text{\hspace{0.17em}⁣}:M_n\to M_n$ preserve Hadamard circulant majorization. In this note, we show that this property is inherited by the three most popular generalized inverses, viz. the Moore-Penrose inverse, the group inverse and the Drazin inverse.