Pre-orders defined by weighted core-EP inverse

Our aim is to introduce new binary relations on the set of all Wg-Drazin invertible bounded linear operator between two Hilbert spaces in terms of the weighted core-EP inverse. Different characterizations of new binary relations are presented as well as block operator matrix forms for operators in these relations. We prove that three of four our new binary relations are pre-orders on the corresponding set and we state as an open problem to check is the fourth relation a pre-order on the same set.