A note on the spectra of a $3\times 3$ operator matrix and application

Abstract

‎In this paper‎, ‎we investigate the essential approximate point‎ ‎spectrum and the essential defect spectrum of a $3\times3$ block‎ ‎operator matrix with unbounded entries and with domain consisting of‎ ‎vectors which satisfy certain relations between their components by‎ ‎means of the Browder resolvent set‎. ‎Furthermore‎, ‎we apply the‎ ‎obtained results to three-Group transport operators in the Banach‎ ‎space $L_{p}( [-a,a]\times[-1,1])$ where $a\gt 0$ and $p \in [1,+\infty).$‎