Perturbations of Fredholm linear relations in Banach spaces with application to 3×33×3-block matrices of linear relations

The main focus in this paper consists of extending the main results of Diagana (2015) to linear relations. Moreover, we apply our obtained results to study some properties of 3×3-block matrices of linear relations in the form, A≔(ABCDEFGHK). Namely, we show that under some suitable conditions on relations B,C,F,D,GB,C,F,D,G and HH, the relation AA is closed if and only if A,EA,E and KK are closed. In addition, we show that if A,EA,E, KK are Fredholm linear relations, then so is the linear relation AA.