The Weyl essential spectrum of a sequence of linear operators in Banach spaces

In the present paper, we introduce and study the convergence of a sequence of closable linear operators in a Banach space. Moreover, we prove that if T-n converges in the generalized sense to T, where T and (T-n)(n is an element of N) are closed linear operators, then there exists a non negative integer element n(0) such that, for all n >= n(0), we have the Weyl essential spectrum of T-n included in the Weyl essential spectrum of T (see Theorem 3.1). The same study is made for the case of convergence to zero compactly under which weaker results are established (Theorem 3.3 and Corollary 3.1). (C) 2015 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.