Integral Equations and Operator Theory

, Volume 44, Issue 3, pp 290–315

Towards a model theory for 2-hyponormal operators

  • Raúl E. Curto
  • Woo Young Lee

DOI: 10.1007/BF01212035

Cite this article as:
Curto, R.E. & Lee, W.Y. Integr equ oper theory (2002) 44: 290. doi:10.1007/BF01212035


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.

2000 Mathematics Subject Classification

Primary 47B2047B3547B37Secondary 47-0447A2047A57

Copyright information

© Birkhäuser Verlag 2002

Authors and Affiliations

  • Raúl E. Curto
    • 1
  • Woo Young Lee
    • 2
  1. 1.Department of MathematicsUniversity of IowaIowa City
  2. 2.Department of MathematicsSungkyunkwan UniversitySuwonKorea