A characterization of k-hyponormality via weak subnormality

Abstract

An operator T acting on a Hilbert space H is said to be weakly subnormal if there exists an extension T ̂ acting on K⊇ H such that T ̂ ∗ T ̂ f= T ̂ T ̂ ∗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]) i,j=1 k 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.