Hyponormal operators with rank-two self-commutators

Abstract

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 a rank-one self-commutator then T is subnormal and if instead T | ker [ T ∗ , T ] has a rank-two self-commutator then T is either a subnormal operator or the kth minimal partially normal extension, T k ˆ ( k ) , of a ( k + 1 ) -hyponormal operator T k 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 + .