The unitary orbit of strongly irreducible operators in the nest algebra with well-ordered set
For a nest algebra \(N\) of operators on a Hilbert space let \(\tau(\mathcal N)\) be the set of all operator \(T\) such that \(TN=N\) for all \(N\in\mathcal N\). An operator \(T\) is strongly irreducible if it does not commute with any nontrivial idempotent. Let \(\Lambda(T)\) denote the spectral picture of \(T\) and \(S(T)^-\) the closure of the similarity orbit of \(T\). The authors prove that if \(\mathcal N\) is maximal and well-ordered and \(T\in\tau(\mathcal N)\) has connected spectrum then there exists a strongly irreducible operator \(A\) in \(\tau(\mathcal N)\) with the same spectral picture as \(T\) and \(T\in S(A)^-\). Moreover, they prove that if \(\mathcal N\) is well ordered with finite dimensional atoms then the closure of the unitary orbit of all strongly irreducible elements of \(\tau(\mathcal N)\) consists of all \(T\) such that its usual and Weyl spectrum are connected and ind\((T-\lambda)\geq 0\) for all \(\lambda\) such that \(T-\lambda\) is semi-Fredholm. As a corollary they prove that, for good domains, if \(T\) is a Cowen-Douglas operator on \(\Omega\) and its spectrum is the closure of \(\Omega\) then for every \(\varepsilon>0\) there is a compact operator \(K\) with \(|K|<\varepsilon\) such that \(T+K\) is strongly irreducible.
scientific article; zbMATH DE number 1046969