Corrigendum an Addendum : “ On the axiomatic theory of spectrum II ”
نویسندگان
چکیده
Recall some notations and terminology from [4]. For closed subspaces M, L of a Banach space X we write M e ⊂L (M is essentially contained in L ) if there exists a finite-dimensional subspace F ⊂ X such that M ⊂ L + F . Equivalently, dim M/(M ∩L) = dim(M + L)/L < ∞. Similarly we write M e =L if M e ⊂L and L e ⊂M . For a (bounded linear) operator T ∈ L(X) write R∞(T ) = ∞n=0 R(T) and N∞(T ) = ⋃∞ n=0 N(T ). An operator T ∈ L(X) is called semi-regular (essentially semi-regular) if R(T ) is closed and N(T ) ⊂ R∞(T ) (N(T ) e ⊂R∞(T ), respectively). Further, T is called quasi-Fredholm if there exists d ≥ 0 such that R(T d+1) is closed and R(T ) + N(T ) = R(T ) + N∞(T ) (equivalently, N(T ) ∩R(T ) = N(T ) ∩R∞(T )). The proof of Theorem 15 of [4] relies on the following statement (where d is an integer whose existence is postulated in the definition of quasi-Fredholm operators): if T is quasi-Fredholm and F of rank 1 then N(T ) ∩R(T ) ⊂ R∞(T + F ).
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