New properties of the core--EP pre-order

Abstract

The first aim of this paper is to present new characterizations of the core--EP pre-order $A\leq^{\tiny\textcircled{\tiny D}}B$ between two Hilbert space operators based on corresponding self-adjoint operators and the powers of the core--EP inverse.

Under the relation $A\leq^{\tiny\textcircled{\tiny D}}B$, we further establish equivalent conditions for the forward order law $(AB)^{\tiny\textcircled{\tiny D}}=A^{\tiny\textcircled{\tiny D}}B^{\tiny\textcircled{\tiny D}}$ to hold. We give conditions for the equivalence between the forward order law $(AB)^{\tiny\textcircled{\tiny D}}=A^{\tiny\textcircled{\tiny D}}B^{\tiny\textcircled{\tiny D}}$ and the reverse order law $(AB)^{\tiny\textcircled{\tiny D}}=B^{\tiny\textcircled{\tiny D}}A^{\tiny\textcircled{\tiny D}}$. Also, in the case that $A\leq^{\tiny\textcircled{\tiny D}}B$, necessary and sufficient conditions for $(B-A)^{\tiny\textcircled{\tiny D}}=B^{\tiny\textcircled{\tiny D}}-A^{\tiny\textcircled{\tiny D}}$ are studied.

Applying our results, we obtain characterizations for the core partial order and

the forward order law for the core inverse.