Further relationships between certain partial orders of matrices and their squares

AbstractTheorem 3 of Baksalary and Pukelsheim [Linear Algebra Appl. 151 (1991) 135] asserts that if both A and B are Hermitian nonnegative definite matrices, then the star order A⩽∗B between them and the star order A2⩽∗B2 between their squares are equivalent and they imply the commutativity property AB=BA. In this paper, relationships between the three conditions mentioned above are reinvestigated in situations where the assumptions on A and B are completely or partially relaxed. Some results concerning the star order are obtained as corollaries to corresponding results referring to the left-star and right-star orders introduced by Baksalary and Mitra [Linear Algebra Appl. 149 (1991) 73]