Investigation on the Hermitian matrix expression‎ ‎subject to some consistent equations

In this paper‎, ‎we study the extremal‎ ‎ranks and inertias of the Hermitian matrix expression $$‎ ‎f(X,Y)=C_{4}-B_{4}Y-(B_{4}Y)^{*}-A_{4}XA_{4}^{*},$$ where $C_{4}$ is‎ ‎Hermitian‎, ‎$*$ denotes the conjugate transpose‎, ‎$X$ and $Y$ satisfy‎ ‎the following consistent system of matrix equations $A_{3}Y=C_{3}‎, ‎A_{1}X=C_{1},XB_{1}=D_{1},A_{2}XA_{2}^{*}=C_{2},X=X^{*}.$ As‎ ‎consequences‎, ‎we get the necessary and sufficient conditions for the‎ ‎above expression $f(X,Y)$ to be (semi) positive‎, ‎(semi) negative‎. ‎The relations between the Hermitian part of the solution to the‎ ‎matrix equation $A_{3}Y=C_{3}$ and the Hermitian solution to the‎ ‎system of matrix equations‎ ‎$A_{1}X=C_{1},XB_{1}=D_{1},A_{2}XA_{2}^{*}=C_{2}$ are also‎ ‎characterized‎. ‎Moreover‎, ‎we give the necessary and sufficient‎ ‎conditions for the solvability to the‎ ‎following system of matrix equations‎ ‎$A_{3}Y=C_{3},A_{1}X=C_{1},XB_{1}=D_{1}‎, ‎A_{2}XA_{2}^{*}=C_{2},X=X^{*}‎, ‎B_{4}Y+(B_{4}Y)^{*}+A_{4}XA_{4}^{*}=C_{4} $ and provide an‎ ‎expression of the general solution to this system‎ ‎when it is solvable‎.