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.

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.