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\section*{Manifold of proper elements} By {\sl S.V. Djordjevic and S. S\'anchez Perales}.  

\medskip  \noindent  Let $X$ be a Banach space and let $B(X)$ denote the space of all bounded linear transformation on $X$. With $$Eig(X)=\{ (\lambda ,L,A)\in \mathbf C\times P_1(X)\times {\mathcal B}(X): A(L)\subset L \mbox{ and } A_{|L}=\lambda I\} $$ we denote the {\it manifold of proper elements of} $X$ and let $(\lambda_0, L_0,A_0)\in Eig (X)$ be a fix arbitrary element.  In the first part of this note we give necessary and sufficient conditions that $(\lambda, L,A)\in Eig (X)$ using the system of equations determinate with $(\lambda_0, L_0,A_0)\in Eig (X)$.  In the second part we apply this result to describe  relation between multiplicity of eigenvalue $\lambda_0$ of the operator $A_0$ and the spectrum of the operator $\widehat{A_0}$ from quotient $X/L_0$ to itself  definite with $\widehat{A_0}(x+L_0)=A_0(x)+L_0$.
%Eigenvalues, Eigenvectors, Multiplicity","15A18","47A10","MS2 Eigenproblems: Theory and computation","14:24:02","Tue Mar 11 2008","148.228.128.56"
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