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Abstract. Let A, B and C be adjointable operators on a Hilbert C * -module E . Giving a suitable version of the celebrated Douglas theorem in the context of Hilbert C * -modules, we present the general solution of the equation AX + Y B = C when the ranges of A, B and C are not necessarily closed. We examine a result of Fillmore and Williams in the setting of Hilbert C * -modules. Moreover, we obtain some necessary and sufficient conditions for existence of a solution for AXA * + BY B * = C. Finally, we deduce that there exist nonzero operators X, Y ≥ 0 and Z such that AXA * + BY B * = CZ, when A, B and C are given subject to some conditions. Introduction and PreliminariesRecently several operator equations have been extended from matrices to infinite dimensional spaces, i.e., Hilbert spaces and Hilbert C * -modules; see [11] and references therein. Recall that the notion of Hilbert C * -module is a natural generalization of that of Hilbert space arising by replacing the field of scalars C by a C * -algebra. Generalized inverses are useful tools for investigation of solutions of operator equations in the setting of Hilbert C * -modules but these inverses need the strong condition of closedness of ranges of considered operators. Fang et al. [6,7] have studied the solvability of operator equations without the closedness condition on ranges of operators by employing a generalization of a known theorem of Douglas [4, Theorem 1] in the framework of Hilbert C * -modules. In their results, concentration is based on the idea of using more general (orthogonal) projections instead of projections such as AA † . They investigated the equations AX = B Inspired by Fang et al., we investigate the solution of equations AX + Y B = C and AXA * +BY B * = C without the condition of closedness of ranges. This paper is organized as follows. First, we recall some basic information about Hilbert C * -modules. In Section 2, we present an example that shows that the conditions CC * ≤ λAA * for some λ > 0 and R(C) ⊆ R(A) are not equivalent in the setting 2010 Mathematics Subject Classification. 15A24, 46L08, 47A05, 47A62.

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Abstract. We investigate positive definite solutions of nonlinear matrix equationswhere Q is a positive definite matrix, Φ and Φ i (1 i m) are positive linear maps on M n (C) and f is a nonnegative matrix monotone or matrix antimonotone function on [0,∞) . In this article, using appropriate inequalities and some fixed point results, we prove the existence of unique positive definite solutions for the mentioned above equations.Mathematics subject classification (2010): 15A24.

Let A, B and C be adjointable operators on a Hilbert C * -module E . Giving a suitable version of the celebrated Douglas theorem in the context of Hilbert C * -modules, we present the general solution of the equation AX + Y B = C when the ranges of A, B and C are not necessarily closed. We examine a result of Fillmore and Williams in the setting of Hilbert C * -modules.Moreover, we obtain some necessary and sufficient conditions for existence of a solution for AXA * + BY B * = C. Finally, we deduce that there exist nonzero operators X, Y ≥ 0 and Z such that AXA * + BY B * = CZ, when A, B and C are given subject to some conditions.