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 T (element of) L(kappa) of T (element of) L(H) we only require that T*T f = TT*f hold for f (element of) H; in this case we call 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 2hyponormal weighted shifts are weakly subnormal. Let (alpha) (identical to)) {(alpha) n } n=0 (infinity) be a weight sequence and let W(alpha) denote the associated unilateral weighted shift on H (identical to) l^2(Z+). If W(alpha) is 2-hyponormal then W(alpha) is weakly subnormal. Moreover, there exists a partially normal extension W(alpha) on (kappa) := H (circled pluse) H such that (i) W(alpha) is hyponormal; (ii) (sigma)(W(alpha)) = (sigma)(W(alpha)); and (iii) ||W(alpha)|| = || W(alpha)||. In particular, if (alpha) is strictly increasing then W(alpha) can be obtained as W(alpha)=(W(alpha) 0 [W(sup *)(sub alpha) , W (alpha)]1/2 W(beta)) on (kappa) := H (circled pluse) H, where W (beta) is a weighted shift whose weight sequence {(beta)n} n=0 (infinity) is given by (beta)n := (alpha)n redical(((alpha)(sub n+1)^2 – (alpha)(sub n)^2)/(alpha)(sub n) ^2 – (alpha)(sub n-1) ^ 2)) (n=0,1,…, (alpha)-1 := 0). In this case, W(alpha) is a minimal partially normal extension of W(alpha) . In addition, if W(alpha) is 3-hyponormal then W(alpha) can be chosen to be weakly subnormal. This allows us to shed new light on Stampfliʹs geometric construction of the minimal normal extension of a subnormal weighted shift. Our methods also yield two additional results: (i) the square of a weakly subnormal operator whose minimal partially normal extension is always hyponormal, and (ii) a 2-hyponormal operator with rank-one self-commutator is necessarily subnormal. Finally, we investigate the connections of weak subnormality and 2-hyponormality with Aglerʹs model theory.