| Summary: | In this paper, using the technique of operator matrix, we consider the positive solution of the system of operator equations <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>X</mi><mo>=</mo><mi>B</mi><mo>,</mo><mi>X</mi><mi>C</mi><mo>=</mo><mi>D</mi></mrow></semantics></math></inline-formula> in the framework of the Hilbert space; here, the ranges <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>(</mo><mi>A</mi><mo>)</mo></mrow></semantics></math></inline-formula> of <i>A</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>(</mo><mi>C</mi><mo>)</mo></mrow></semantics></math></inline-formula> of <i>C</i> are not necessarily closed. Firstly, we provide a new necessary and sufficient condition for the existence of positive solutions of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>X</mi><mo>=</mo><mi>B</mi></mrow></semantics></math></inline-formula> and also provide a representation of positive solutions, which generalize previous conclusions. Furthermore, using the above result, a condition of equivalence for the existence of common positive solutions of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>X</mi><mo>=</mo><mi>B</mi><mo>,</mo><mi>X</mi><mi>C</mi><mo>=</mo><mi>D</mi></mrow></semantics></math></inline-formula> is given, as well as the general forms of positive solutions.
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