New Trends into 3 × 3 Blocks of Operator Matrices Modeled with Radiative Transfer Equations

The radiative transfer equation is one of the most fundamental relationships for describing macroscopic thermal radiation transfer processes. Based on this, the main contribution of this paper is to provide new trends into an unbounded block 3×3 operator matrix defined with a non-maximal domain, offering new insights into the phenomena of radiative transport equations. More precisely, we formulate sufficient and necessary conditions for the entries of the above mentioned model: of the operator matrix to provide an explicit formula for its associated Frobenius–Schur decomposition. Some new spectral properties of such matrix model are also derived to relate its eigenvalues to their two associated Schur complements. As an illustrative example of our findings, we state a model of 3×3 groups of radiative transfer equations in a channel with specific boundary conditions. Our findings build upon and enhance the recent advancements of Bouzidi et al. (Georgian Mathematical Journal, 32, 1, (2025), 37–50). In particular, this study expands the theoretical scope to the realm of 3×3 operator matrix form with a non-maximum domain.