New approach on the study of operator matrix

In the present paper, a new technique is presented to study the problem of invertibility of unbounded block 3 x 3 operator matrices defined with diagonal domain. Sufficient criteria are established to guarantee our interest and to prove some interaction between such a model of an operator matrix and its diagonal operator entries. The effectiveness of the proposed new technique is shown by a physical example of an integro differential equation named the neutron transport equation with partly elastic collision operators. In particular, the obtained results answer the question in [H. Zguitti, A note on Drazin invertibility for upper triangular block operators, Mediterr. J. Math. 10 (2013), no. 3, 1497-1507] and the conjecture in [A. Bahloul and I. Walha, General- ized Drazin invertibility of operator matrices, Numer. Funct. Anal. Optim. 43 (2022), no. 16, 1836-1847].