This paper deals with a new model of unbounded block \(2\times 2\) operator matrix originated by V. Casarino et al. in Casarino et al. (Integr Equ Oper Theory 47:289–306, 2003), so called the one sided operator matrix form, in order to investigate some advances in the spectral analysis of such model of operator matrix. Our approach allows us to define an abstract setting to treat some essential spectra of one sided operator matrix form. Sufficient conditions in terms of the operators entries of such model of operator matrix have been established to guarantee our interest and to prove some progress in the theory of operators matrices and to provide an amelioration of many earlier works. Precisely, our contributions in this paper aim to give a positive answer to the open question proposed by F. Abdmouleh et al. in Abdmouleh et al. (Georgian Math J 30:161–172, 2023). An example of two groups of Radiative transfer equations in a channel with periodic boundary conditions is tested to ensure the validity of our abstract results.
