The Proper Elements and Simple Invariant Subspaces

Abstract

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)\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 $\lambda_0$ is a simple pole of $A_0$ if and only if $\lambda_0\notin\sigma (\widehat{A_0})$. Follow this concept we can define simple invariant subspaces of linear operator $T$ like invariant subspace $E$ such that $\sigma (T_E)\cap\sigma (\widehat{T_E})=\emptyset$, where $T_{E}:E\to E$ is the restriction of $T$ on $E$, $\widehat{T_E}$ is the  operator $\widehat{T_E}(\pi (y))=\pi (T(y))$ 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.