Small compact perturbation of strongly irreducible operators. (Q1865934)

Let \(\mathcal H\) be a complex, separable Hilbert space of infinite dimension. An element \(T\) in \(\mathcal L(\mathcal H)\), the algebra of all linear and bounded operators in \(\mathcal H\), is called strongly irreducible, if it is not similar to any reducible operator. It is easy to see that for any compact \(K\in \mathcal L(\mathcal H)\) the operator \(T+K\) is not strongly irreducible provided the spectrum of \(T\) is not connected. In this article, it is shown that for any \(T\in\mathcal L(\mathcal H)\) with connected spectrum there is a compact \(K\in \mathcal L(\mathcal H)\) of arbitrarily small norm such that \(T+K\) is strongly irreducible. In the second main result, it is assumed that \(T\) belongs to the \(n\)-th Cowen-Douglas complex \(\mathcal B_n(\Omega)\) for some open, bounded and connected subset \(\Omega\) of the complex plane. The authors prove that in this case \(K\) may be choosen such that \(T+K\) also belongs to \(\mathcal B_n(\Omega)\).