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Solutions to Operator Equations on Hilbert C*-Modules II

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Abstract

In this paper, we study the solvability of the operator equations A*X + X*A = C and A*XB + B*X*A = C for general adjointable operators on Hilbert C*-modules whose ranges may not be closed. Based on these results we discuss the solution to the operator equation AXB = C, and obtain some necessary and sufficient conditions for the existence of a real positive solution, of a solution X with B*(X* + X)B ≥ 0, and of a solution X with B*XB ≥ 0. Furthermore in the special case that \({R(B)\subseteq\overline{R(A*)}}\) we obtain a necessary and sufficient condition for the existence of a positive solution to the equation AXB = C. The above results generalize some recent results concerning the equations for operators with closed ranges.

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Authors and Affiliations

  1. Department of Mathematics, Tongji University, 200092, Shanghai, China

    Xiaochun Fang & Jing Yu

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  1. Xiaochun Fang
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  2. Jing Yu
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Correspondence to Xiaochun Fang.

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The research reported in this article was supported by the National Natural Science Foundation of China (10771161).

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Fang, X., Yu, J. Solutions to Operator Equations on Hilbert C*-Modules II. Integr. Equ. Oper. Theory 68, 23–60 (2010). https://doi.org/10.1007/s00020-010-1783-x

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  • DOI: https://doi.org/10.1007/s00020-010-1783-x

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