Common hermitian and positive solutions to the adjointable operator equations AX=C, XB=D Original Research Article

Let image be a C*-algebra. For any Hilbert image-modules H and K, let image be the set of adjointable operators from H to K. Let H,K,L be Hilbert image-modules, image and image. In this paper, we propose necessary and sufficient conditions for the existence of common hermitian and positive solutions image to the equations image, and obtain the formulae for the general forms of these solutions. Some results, known for finite matrices and Hilbert space operators, are extended to the adjointable operators acting on Hilbert C*-modules.