The closure of the unitary orbit of the set of strongly irreducible operators in non-well ordered nest algebra

摘要

A bounded linear operator T on a Hilbert space H is strongly irreducible if T does not commute with any non-trivial idempotent. A nest N is a chain of subspaces of H contain {0} and H, which is closed under intersection and closed span. The nest algebra alg N associated with N is the set of all operators which leave each subspace in N invariant. This paper proves that the norm closure of the unitary orbit of the strongly irreducible operators in a nest algebra is the set of operators whose spectrum is connected if and only if N or N~⊥are not well-ordered.