The Proper Elements and Simple Invariant Subspaces - Djordjević, Slaviša V. (Communications in Mathematics and Applications, 2012)
A proper element of X is a triple (λ, L, A) composed by an eigenvalue λ, an invariant subspace of an operator A in B(X) generated by one eigenvector of λ and the operator A. For (λ_0, L_0,A_0)Eig (X), where L_0=L ({x_0} ), the operator A_0 induces an operator A_0 from the quotient X/L_0 into itself, i.e. A_0(x+L_0)=A_0(x)+L_0. In paper we show that λ_0 is a simple pole of A_0 if and only if λ_0 otinσ (A_0). Follow this concept we can define simple invariant subspaces of linear operator T like invariant subspace E such that σ (T_E)σ (T_E)=, where T_E:E→ E is the restriction of T on E, T_E is the operator T_E(π (y))=π (T(y)) on the quotient space X/E and π is the natural homoeomorphism between X and X/E. Also, we give some properties of stability of simple invariant subspaces.