2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73522We generalize our earlier results from \cite{K} on the Bessmertny\uı class of operator-valued functions holomorphic in the open right poly-halfplane which admit representation as a Schur complement of a block of a linear homogeneous operator-valued function with positive semidefinite operator coefficients, to the case of a product of open right matrix halfplanes. Several equivalent characterizations of this generalized Bessmertny\uı class are presented. In particular, its intimate connection with the Agler--Schur class of holomorphic contractive operator-valued functions on the product of matrix unit disks is established.Functional Analysis47A48; 32A10; 47A56; 47A60text