On Analytical Solutions of Operator Equation Constructed by The Adjointable Operator

Recently, Operator Equation Theory (OET) has a leading demonstrated potentiality applicable in numerous scientific ranges of engineering, physical and mathematical. In a Hilbert C*-module, OET has enhanced by expanding upon extensive research. In this study, for the general situation of adjointable operators, the solvability of the operator equation P^* XΦ^*+ΦYP=Ω, where X and Y are unknown operators, are investigated based on Moore-Penrose inverse. Necessary and sufficient conditions for founding a solution to this equation are proposed. Moreover, by utilizing matrix approaches, four general expressions for the solutions are derived depending on the states of the operators P and Φ involved in the equation.