The authors characterize the (U+K)-orbits of a class essentially normal operators and prove that some essentially normal operators with connected spectrum are strongly irreducible after a small compact perturbation. This partially answers a question of Domigo A. Herrero.

A bounded linear operator T on a Hdbert 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 i s 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 N1 are not well-ordered. Strongly irreducible operator, nest, nest algebra, unitary orbit, spectrum.

Let T be a bounded linear operator acting on a separable infinite dimensional Hilbert space. Let E be a positive number. In this article, we prove that the perturbation of T by a compact operator K with ‖K‖<E can be strongly irreducible if T is a quasitriangular operator with the spectrum σ(T) connected. The Main Theorem of this article nearly answers the question below posed by D. A. Herrero. Suppose that T is a bounded linear operator acting on a separable infinite dimensional Hilbert space with σ(T) connected. Let ε>0 be given. Is there a compact operator K with ‖K‖<ε such that T+K is strongly irreducible?

It is characterized when a bilateral operator weighted shift is a Cowen-Douglas operator.