In this paper we study proper holomorphic maps between bounded symmetric domains when the source domain is not irreducible. More precisely, we provide sufficient conditions for semi-product proper holomorphic maps to be product proper. As an application we characterize proper holomorphic maps between equidimensional bounded symmetric domains.
"Remark on Proper Holomorphic Maps Between Reducible Bounded Symmetric Domains." Taiwanese J. Math. 22 (2) 325 - 337, April, 2018. https://doi.org/10.11650/tjm/170806