# Towards a model theory for 2-hyponormal operators

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## Abstract

We introduce the notion of where In this case,\(\widehat{W}_\alpha \) is a minimal partially normal extension of

*weak subnormality*, which generalizes subnormality in the sense that for the extension\(\widehat{T}\) ∈\(\mathcal{L}(\mathcal{K})\) of*T*∈\(\mathcal{L}(\mathcal{H})\) we only require that\(\widehat{T}^* \widehat{T}f = \widehat{T}\widehat{T}^* f\) hold for*f*∈\(\mathcal{H}\); in this case we call\(\widehat{T}\) 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 ≡ {α_{n}}_{n=0}^{∞}be a weight sequence and let*W*_{α}denote the associated unilateral weighted shift on\(\mathcal{H} \equiv \ell ^2 (\mathbb{Z}_ +)\). If*W*_{α}is 2-hyponormal then*W*_{α}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$$\widehat{W}_\alpha = \left( {\begin{array}{*{20}c} {W_\alpha } \\ 0 \\ \end{array} \begin{array}{*{20}c} {[W_\alpha ^* ,W_\alpha ]^{\frac{1}{2}} } \\ {W_\beta } \\ \end{array} } \right)on\mathcal{K}: = \mathcal{H} \oplus \mathcal{H},$$

*W*_{β}is a weighted shift whose weight sequence {β_{n}·_{n=0}^{∞}is given by$$\beta _n : = \alpha _n \sqrt {\frac{{\alpha _{n + 1}^2 - \alpha _n^2 }}{{\alpha _n^2 - \alpha _{n - 1}^2 }}} (n = 0,1,...;\alpha - 1: = 0).$$

*W*_{α}. In addition, if*W*_{α}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.### 2000 Mathematics Subject Classification

Primary 47B20 47B35 47B37 Secondary 47-04 47A20 47A57## Preview

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© Birkhäuser Verlag 2002