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We review some of the significant generalizations and applications of thecelebrated Douglas theorem on the equivalence of factorization, rangeinclusion, and majorization of operators. We then apply it to find acharacterization of the positivity of $2\times 2$ block matrices of operatorsin Hilbert spaces and finally describe the nature of such block matrices andprovide several ways for showing their positivity.

Positivity of $2\times 2$ block matrices of operators