We review some significant generalizations and applications of the celebrated Douglas theorem on equivalence of factorization, range inclusion, and majorization of operators. We then apply it to find a characterization of the positivity of block matrices of operators on Hilbert spaces. Finally, we describe the nature of such block matrices and provide several ways for showing their positivity.
"Positivity of block matrices of operators." Banach J. Math. Anal. 13 (3) 726 - 743, July 2019. https://doi.org/10.1215/17358787-2019-0019