Solutions To Operator Equations Over Hilbert C~*-modules

Hilbert C*-modules are generalizations of Hilbert spaces by allowing inner products to take values in some C*-algebra instead of complex numbers.Solutions of operator equations have always been a hot topic,and they have also been extended to Hilbert C*-modulesLet T,S and T'i(i=1,2)be adjointable operators on a Hilbert C*-modules E.We obtain some necessary and sufficient conditions for the existence of common solutions for the system of operator equations TX=T'1 and TXS=T'2 when the ranges of T,S and T'i(i=1,2)are not necessarily closed,and give a general form of common solutions Moreover we investigate the solutions of operators equation TXS+SYT=T' when the ranges of T and S are not not necessarily closed.The above results generalize some recent results concerning the equations for operators with closed rangesLet A1,A2 ?(?)(E)and C be adjointable operators on a Hilbert C*-modules E.We obtain some necessary and sufficient conditions for the existence of positive solutions for the equations A1X1A1*+A2X2A2=C when the considered operators are regular,and give a general form of positive solutions.Moreover we investigate the necessary and sufficient conditions for the exitence of common positive solutions of the system of operators equa-tions A1X1=C1,X1B2=C2,A3X2=C3,X3B4=C4 and A5X1A5*+A6X2A6*=C5 when the considered operators are regular.The above results generalize some recent results concerning the positive solutions of operators equations on Hilbert C*-modules.