On Hypo-Jordan operators

In this paper, we show that if T = S + N, where S is similar to a hyponormal operator, S and N commute and N is a nilpotent operator of order m (i.e., N^m = 0), then T is a subscalar operator of order 2m. As a corollary, we get that such a T has a nontrivial invariant subspace if its spectrum σ(T) has the property that there exists some non-empty open set U such that σ(T) ∩ U is dominating for U.