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)\in Eig (X)$, where $L_0=\mathcal{L} (\{x_0\} )$, the operator $A_0$ induces an operator $\widehat{A_0}$ from the quotient $X/L_0$ into itself, i.e. $\widehat{A_0}(x+L_0)=A_0(x)+L_0$. In paper we show that $λ_0$ is a simple pole of $A_0$ if and only if $λ_0 otinσ (\widehat{A_0})$. Follow this concept we can define simple invariant subspaces of linear operator $T$ like invariant subspace $E$ such that $σ (T_E)\capσ (\widehat{T_E})=\emptyset$, where $T_{E}:E→ E$ is the restriction of $T$ on $E$, $\widehat{T_E}$ is the operator $\widehat{T_E}(π (y))=π (T(y))$ 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.
Keywords: Eigenvalues; Eigenvectors; Invariant subspaces