An operator T on H is called strongly irreducible if T does not commute with any nontrivial idernpotent operator In this paper, we first show that each nest algebra T(N) has strong y rreducible operators, Secondly. we obtain a characterization of operators which can be uniquely written as a direct sum of finitely many strongly irreducible operators. Finafiy, we characterize the strongly irreducibility of operators in a nest algebra T(N).

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.

In this paper. N always denotes continuous nest, and T(N)={TεL(H), TM с M; M ε N}. We describe the spectral picture of some strongly irreducible operators in T(N), and characterize the closure of the unitary orbit of the strongly irreducible operaiors in T(N). Continuous nest, Nest algebra, Strongly irreducible, Spectral picture

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?