Quasitriangular plus small compact = strongly irreducible

Let T be a bounded linear operator acting on a separable infinite dimension al Hilbert space. Let epsilon be a positive number. In this article, we pro ve that the perturbation of T by a compact operator K with parallel to K pa rallel to < epsilon can be strongly irreducible if T is a quasitriangular o perator with the spectrum sigma(T) connected. The Main Theorem of this arti cle 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 sigma(T) connected. Let epsilon > 0 be given . Is there a compact operator K with parallel to K parallel to < epsilon su ch that T + K is strongly irreducible?