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 TnTn converges in the generalized sense to TT, where TT and (Tn)n∈N(Tn)n∈N are closed linear operators, then there exists a non negative integer element n0n0 such that, for all n≥n0n≥n0, we have the Weyl essential spectrum of TnTn included in the Weyl essential spectrum of TT (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).