Towards a Model Theory for 2–hyponormal Operators

We introduce the notion of weak subnormality, which generalizes subnormality in the sense that for the extension b T ∈ L(K) of T ∈ L(H) we only require that b T ∗ b Tf = b T b T ∗f hold for f ∈ H; in this case we call b T a partially normal extension of T . After establishing some basic results about weak subnormality (including those dealing with the notion of minimal partially normal extension), we proceed to characterize weak subnormality for weighted shifts and to prove that 2-hyponormal weighted shifts are weakly subnormal. Let α ≡ {αn}n=0 be a weight sequence and let Wα denote the associated unilateral weighted shift on H ≡ `2(Z+). If Wα is 2-hyponormal then Wα is weakly subnormal. Moreover, there exists a partially normal extension c Wα on K := H⊕H such that (i) c Wα is hyponormal; (ii) σ(c Wα) = σ(Wα); and (iii) ||c Wα|| = ||Wα||. In particular, if α is strictly increasing then c Wα can be obtained as c Wα = Wα [W ∗ α,Wα] 2 0 Wβ ! on K := H⊕H,