Positive Solutions of a Class of Operator Equations

Positive solutions of a class of matrix equations were studied by Bhatia, et al., Bull. London Math. Soc., 32, 214 (2000), SIAM J. Matrix Anal. Appl., 14, 132 (1993) and 27, 103–114 (2005), by Kwong, Linear Algebra Appl., 108, 177–197 (1988), and by Cvetkovic and Milovanovic, [Linear Algebra Appl., 429, 2401–2414 (2008)]. Following the idea used in the last paper, we study a class of operator equations in infinite-dimensional spaces and prove that the positivity of solutions can be established for this class of equations under the condition that a certain rational function is positive semidefinite.