Abstract :
An operator T acting on a Hilbert space H is said to be weakly subnormal if there exists an
extension Tˆ acting on K ⊇H such that Tˆ∗Tˆf = TˆTˆ∗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 ,T i ])k
i,j=1 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 T A
0 B 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.
2003 Elsevier Science (USA). All rights reserved
