Limits of strongly irreducible operators, and the Riesz decomposition theorem (Q810842)

Let \({\mathcal L}({\mathcal H})\) be the algebra of all bounded linear operators acting on a complex, separable, infinite dimensional Hilbert space \({\mathcal H}\). The class of strongly irreducible operators is \[ {\mathcal S}{\mathcal J}({\mathcal H})=\{T\in {\mathcal L}({\mathcal H}):\;T\quad does\quad not\quad commute\text{ with } any\quad nontrivial\quad idempotent\}. \] The authors prove that the norm closure of \({\mathcal S}{\mathcal J}({\mathcal H})\) is \(\{\) \(T\in {\mathcal L}({\mathcal H}):\sigma\) (T) is connected\(\}\). Here \(\sigma\) (T) is the spectrum of T. This theorem relates to the Riesz decomposition theorem which says that if \(\sigma\) (T) is not connected, T does commute with a nontrivial idempotent operator. The proof requires some careful constructions of operators which satisfy certain spectral properties.