On the criteria of a measure of non-strict cosingularity in the description of spectral properties of operator matrix

Abstract In this paper, we are interested to formulate new assumptions on the entries of an unbounded 3 × 3 3\times 3 block operator matrix defined with a maximal domain on the product of Banach spaces guaranteeing its corresponding Frobenius–Schur formula. Our approach allows us to derive some original stability results intervening in the theory of perturbed lower semi-Fredholm operators involving the concept of a measure of non-strict cosingularity perturbation. A new technique is presented to investigate the Weidmann and defect essential spectra of the closure of such model of operator matrix via new criterion of perturbation.