An operator T on H is called strongly irreducible if T is not similar to any reducible operators, in this paper, we shall say 'yes' to answer the following question raised hy D. A. Herrero. G yen an operator T with connecteg spectrum a(T) and a positive number E, can we find a compact operator K with ‖K‖＜E such that T+K is strong y irreducible?

For the bounded linear operators acting on a complex separable Hilbert space H, we introduce a binary relation ～sas called strongly approxi-mative similarity It lies between the similarity and the essential similarity. For a class of biquasitriangular operators and a class of quasitriangular operators, this relation is characterized respectively. As a result, the relation ～sas is an equivalent relation in this two cases.

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.