The Proper Elements and Simple Invariant Subspaces

A proper element of X is a triple (lambda, L, A) composed by an eigenvalue lambda, an invariant subspace of an operator A in B(X) generated by one eigenvector of lambda and the operator A. For (lambda(0), L-0, A(0)) is an element of Eig(X), where L-0 = L({x(0)}), the operator Ao induces an operator Ao from the quotient X/Lo into itself, i.e. (A(0)) over cap (x + L-0) = A(0)(x)+L-0. In paper we show that lambda(0) is a simple pole of Ao if and only if lambda(0) is not an element of sigma<((A(0)))over cap>. Follow this concept we can define simple invariant subspaces of linear operator T like invariant subspace E such that sigma(T-E) boolean AND sigma<((T-E))over cap> = phi, where T-E : E -> E is the restriction of T on E, (T-E) over cap is the operator (T-E) over cap (pi(y)) = pi(T(T(y)) on the quotient space X/E and pi is the natural homoeomorphism between X and X/E. Also, we give some properties of stability of simple invariant subspaces.