On the cellular indecomposable property of semi-Fredholm operators (Q1938729)

Let \(H\) be an infinite-dimensional Hilbert space and \(T\) be a linear bounded operator on \(H\). Denote by \(\rho_{sF}(T)\) the set of all complex numbers \(\lambda\) such that \(T-\lambda I\) is semi-Fredholm. A point \(\lambda\in\rho_{sF}(T)\) is called \textit{singular} if the projections \(P_{\ker(T-\lambda I)}\) do not converge to \(P_{\ker(T-\lambda_{0} I)}\) as \(\lambda\rightarrow\lambda_{0}\) in the strong operator topology. An operator \(T\) is said to have the \textit{cellular indecomposable property} if any two of its nontrivial invariant subspaces possess a nontrivial intersection. The main result of the paper says that, if \(T\) is cellular indecomposable, then \(\rho_{sF}(T)\) contains no singular points.