Towards a model theory for 2-hyponormal operators

Abstract

We introduce the notion ofweak subnormality, which generalizes subnormality in the sense that for the extension\(\widehat{T}\) ∈\(\mathcal{L}(\mathcal{K})\) ofT ∈\(\mathcal{L}(\mathcal{H})\) we only require that\(\widehat{T}^* \widehat{T}f = \widehat{T}\widehat{T}^* f\) hold forf ∈\(\mathcal{H}\); in this case we call\(\widehat{T}\) a partially normal extension ofT. 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 letWα denote the associated unilateral weighted shift on\(\mathcal{H} \equiv \ell ^2 (\mathbb{Z}_ +)\). IfWα is 2-hyponormal thenWα is weakly subnormal. Moreover, there exists a partially normal extension\(\widehat{W}_\alpha\) on\(\mathcal{K}: = \mathcal{H} \oplus \mathcal{H}\) such that (i)\(\widehat{W}_\alpha\) is hyponormal; (ii)\(\sigma (\widehat{W}_\alpha) = \sigma (W_\alpha)\); and (iii)\(\parallel \widehat{W}_\alpha \parallel = \parallel W_\alpha \parallel \). In particular, if α is strictly increasing then\(\widehat{W}_\alpha\) can be obtained as $$\widehat{W}_\alpha = \left( {\begin{array}{*{20}c} {W_\alpha } \\ 0 \\ \end{array} \begin{array}{*{20}c} {[W_\alpha ^* ,W_\alpha ]^{\frac{1}{2}} } \\ {W_\beta } \\ \end{array} } \right)on\mathcal{K}: = \mathcal{H} \oplus \mathcal{H},$$ whereWβ is a weighted shift whose weight sequence {βn·n=0∞ is given by $$\beta _n : = \alpha _n \sqrt {\frac{{\alpha _{n + 1}^2 - \alpha _n^2 }}{{\alpha _n^2 - \alpha _{n - 1}^2 }}} (n = 0,1,...;\alpha - 1: = 0).$$ In this case,\(\widehat{W}_\alpha \) is a minimal partially normal extension ofWα. In addition, ifWα is 3-hyponormal then\(\widehat{W}_\alpha\) can be chosen to be weakly subnormal. This allows us to shed new light on Stampfli's 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 Agler's model theory.