Positive and re-positive solutions to some systems of adjointable operator equations over Hilbert C*-modules
Guang Song, Qing‐Wen Wang · Electronic Journal of Linear Algebra · 2011
Abstract. A necessary and sufficient condition for the existence of the general common positive solution to equations A1X = C1, XB2 = C2, A3XA ∗ 3 = C3, A4XA ∗ 4 = C4 for operators between Hilbert C ∗-modules is established, and an expression for the common positive solution to the equations is derived when the solvability conditions are satisfied. As an application, a new necessary and sufficient condition for the system of adjointable operator equations AX = C, XB = D over Hilbert C ∗-modules to have a common Re-positive solution is proved. Moreover, an expression of the general Re-positive solution is derived when the consistent conditions are met. The results of this paper extend some known results in the literature.