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?