1 Manifold of proper elements

By S.V. Djordjevic and S. Snchez Perales.

Let X be a Banach space and let B(X) denote the space of all bounded linear transformation on X. With

Eig(X) = {(λ, L, A) ∈ C × P_{1}(X) × B(X) : A(L) ⊂ L and A_{|L} = λI}

we denote the manifold of proper elements of X and let (λ_{0}, L_{0}, A_{0}) ∈ Eig(X) be a fix arbitrary element. In the first part of this note we give necessary and sufficient conditions that (λ, L, A) ∈ Eig(X) using the system of equations determinate with (λ_{0}, L_{0}, A_{0}) ∈ Eig(X). In the second part we apply this result to describe relation between multiplicity of eigenvalue λ_{0} of the operator A_{0} and the spectrum of the operator Â_{0} from quotient X/L_{0} to itself definite with Â_{0}(x + L_{0}) = A_{0}(x) + L_{0}.