FACTA UNIVERSITATIS (NI ˇ S) Ser. Math. Inform. 17 (2002), 35–55 ON THE SCHECHTER ESSENTIAL SPECTRUM ON BANACH SPACES AND APPLICATION Aref Jeribi Abstract. This paper is devoted to the investigation of the stability of the Schechter essential spectrum of closed densely defined linear operators A sub- jected to additive perturbations K such that (λ - A - K) -1 K or K(λ - A - K) -1 belonging to arbitrary subsets of L(X) (where X denotes a Banach spaces) con- tained in the set J (X). Our approach consists principally in considering the class of A-closable (not necessarily bounded) which contained in the set of A- resolvent Fredholm perturbations which zero index (see Definition 3.5). They are used to describe the Schechter essential spectrum of singular neutron transport equations in bounded geometries. 1. Introduction Let X and Y be two Banach spaces. By an operator A from X into Y we mean a linear operator with domain D(A) ⊂ X and range R(A) ⊂ Y . We denote by C (X,Y ) (resp. L(X,Y )) the set of all closed, densely defined linear operators (resp. the Banach algebra of all bounded linear operators) from X into Y . The subset of all compact operators of L(X,Y ) is designated by K(X,Y ). If X = Y then L(X,Y ), K(X,Y ) and C (X,Y ) are replaced, respectively, by L(X ), K(X ) and C (X ). Definition 1.1. An operator A ∈L(X,Y ) is said to be weakly compact if A(B) is relatively weakly compact in Y for every bounded subset B ⊂ X . The family of weakly compact operators from X into Y is denoted by W(X,Y ). If X = Y the family of weakly compact operators on X , W(X ):= W(X,X ), is a closed two-sided ideal of L(X ) containing K(X ) (cf. [4, 6]). Received May 29, 2001. 2000 Mathematics Subject Classification. Primary 47A10. 35