A characterization of k-hyponormality via weak subnormality

AbstractAn operator T acting on a Hilbert space H is said to be weakly subnormal if there exists an extension T̂ acting on K⊇H such that T̂∗T̂f=T̂T̂∗f for all f∈H. When such partially normal extensions exist, we denote by m.p.n.e.(T) the minimal one. On the other hand, for k⩾1, T is said to be k-hyponormal if the operator matrix ([T∗j,Ti])i,j=1k is positive. We prove that a 2-hyponormal operator T always satisfies the inequality T∗[T∗,T]T⩽‖T‖2[T∗,T], and as a result T is automatically weakly subnormal. Thus, a hyponormal operator T is 2-hyponormal if and only if there exists B such that BA∗=A∗T and TA0B is hyponormal, where A:=[T∗,T]1/2. More generally, we prove that T is (k+1)-hyponormal if and and only if T is weakly subnormal and m.p.n.e.(T) is k-hyponormal. As an application, we obtain a matricial representation of the minimal normal extension of a subnormal operator as a block staircase matrix