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)

kerT^∗,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_|kerT^∗,T] has a rank-one self-commutator then T is subnormal and if instead T_|kerT^∗,T] has a rank-two self-commutator then T is either a subnormal operator or the kth minimal partially normal extension, , of a (k+1)-hyponormal operator Tk which has a rank-two self-commutator for any k∈Z₊. Hence, in particular, every weakly subnormal (or 2-hyponormal) operator with a rank-two self-commutator is either a subnormal operator or a finite rank perturbation of a k-hyponormal operator for any k∈Z₊.