Hyponormal Operators with Rank-two Self-commutators

In this paper it is shown that if T ∈ L(H) satisfies (i) T is a pure hyponormal operator; (ii) [T ∗, T ] is of rank-two; and (iii) ker [T ∗, T ] is invariant for T , then T is either a subnormal operator or the Putinar’s matricial model of rank two. More precisely, if T |ker [T∗,T ] has the rank-one self-commutator then T is subnormal and if instead T |ker [T∗,T ] has the ranktwo self-commutator then T is either a subnormal operator or the k-th minimal partially normal extension, T̂k (k) , of a (k+1)-hyponormal operator Tk which has rank-two self-commutator for any k ∈ Z+. Hence, in particular, every weakly subnormal (or 2-hyponormal) operator with rank-two self-commutator is either a subnormal operator or a finite rank-perturbation of a k-hyponormal operator for any k ∈ Z+.