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 ≡ {α n } ∞ n=0 be a weight sequence and letW α denote the associated unilateral weighted shift on\(\mathcal{H} \equiv \ell ^2 (\mathbb{Z}_ +)\). IfW α is 2-hyponormal thenW 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
whereW β is a weighted shift whose weight sequence {β n · ∞ n=0 is given by
In this case,\(\widehat{W}_\alpha \) is a minimal partially normal extension ofW α. In addition, ifW 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.
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References
J. Agler,An abstract approach to model theory, Surveys of Some Recent Results in Operator Theory, II, Pitman Research Notes in Mathematics, Vol192, John Wiley and Sons, New York, 1988, pp. (1–23).
A. Aleman,Subnormal operators with compact selfcommutator, Manuscripta Math.91 (1996), 353–367.
A. Athavale,On joint hyponormality of operators, Proc. Amer. Math. Soc.103 (1988), 417–423.
J. Bram,Subnormal operators, Duke Math. J.22 (1955), 75–94.
Y.B. Choi,A propagation of quadratically hyponormal weighted shifts, Bull. Korean Math. Soc.37 (2000), 347–352.
J.B. Conway,The Theory of Subnormal Operators, Math. Surveys and Monographs, Vol.36, Amer. Math. Soc., Providence, 1991.
J.B. Conway and W. Szymanski,Linear combination of hyponormal operators, Rocky Mountain J. Math.18 (1988), 695–705.
C. Cowen,Hyponormal and subnormal Toeplitz operators, Surveys of Some Recent Results in Operator Theory, I (J.B. Conway and B.B. Morrel, eds.), Pitman Research Notes in Mathematics, Vol171, Longman, 1988, pp. (155–167).
C.C. Cowen and J.J. Long,Some subnormal Toeplitz operators, J. Reine Angew. Math.351 (1984), 216–220.
R.E. Curto,Quadratically hyponormal weighted shifts, Integral Equations Operator Theory13 (1990), 49–66.
—,Joint hyponormality: A bridge between hyponormality and subnormality, Operator Theory: Operator Algebras and Applications (Durham, NH, 1988) (W.B. Arveson and R.G. Douglas, eds.), Proc. Sympos. Pure Math., Vol.51, part II, American Mathematical Society, Providence, (1990), Part 11, 69–91.
R.E. Curto and L.A. Fialkow,Recursiveness, positivity, and truncated moment problems, Houston J. Math.17 (1991), 603–635.
—,Recursively generated weighted shifts and the subnormal completion problem, Integral Equations Operator Theory17 (1993), 202–246.
—,Recursively generated weighted shifts and the subnormal completion problem, II, Integral Equations Operator Theory18 (1994), 369–426.
R.E. Curto, I.B. Jung and W.Y. Lee,Extensions and extremality of recursively generated weighted shifts, Proc. Amer. Math. Soc.130 (2002), 565–576.
R.E. Curto and W.Y. Lee,Joint hyponormality of Toeplitz pairs, Mem. Amer. Math. Soc. no.712, Amer. Math. Soc., Providence, 2001.
R.E. Curto,Flatness, perturbations and completions of weighted shifts (preprint 2000).
R.E. Curto, P.S. Muhly and J. Xia,Hyponormal pairs of commuting operators, Contributions to Operator Theory and Its Applications (Mesa,AZ, 1987) (I. Gohberg, J.W. Helton and L. Rodman, eds.), Operator Theory: Advances and Applications, vol.35, Birkhäuser, Basel-Boston, (1988), 1–22.
R.E. Curto and M. Putinar,Existence of non-subnormal polynomially hyponormal operators, Bull. Amer. Math. Soc. (N.S.)25 (1991), 373–378.
—,Nearly subnormal operators and moment problems, J. Funct. Anal.115 (1993), 480–497.
R.G. Douglas, V.I. Paulsen, and K. Yan,Operator theory and algebraic geometry, Bull. Amer. Math. Soc. (N.S.)20 (1989), 67–71.
M.A. Dritschel and S. McCullough,Model theory for hyponormal contractions, Integral Equations Operator Theory36 (2000), 182–192.
P. Fan,A note on hyponormal weighted shifts, Proc. Amer. Math. Soc.92 (1984), 271–272.
P.R. Halmos,Ten problems in Hilbert space, Bull. Amer. Math. Soc.76 (1970), 887–933.
—,A Hilbert Space Problem Book, 2nd ed., Springer, New York, 1982.
J.K. Han, H.Y. Lee and W.Y. Lee,Invertible completions of 2×2 upper triangular operator matrices, Proc. Amer. Math. Soc.128 (2000), 119–123.
A. Joshi,Hyponormal polynomials of monotone shifts, Indian J. Pure Appl. Math.6 (1975), 681–686.
G. Lumer and M. Rosenblum,Linear operator equations, Proc. Amer. Math. Soc.10 (1959), 32–41.
J.E. McCarthy and L. Yang,Subnormal operators and quadrature domains, Adv. Math.127 (1997), 52–72.
S. McCullough and V. Paulsen,A note on joint hyponormality, Proc. Amer. Math. Soc.107 (1989), 187–195.
B.B. Morrel,A decomposition for some operators, Indiana Univ. Math. J.23 (1973), 497–511.
R.F. Olin, J.E. Thomson and T.T. Trent,Subnormal operators with finite rank self-commutator (preprint 1990).
A. Shields,Weighted shift operators and analytic function theory, Math. Surveys13 (1974), 49–128.
J. Stampfli,Which weighted shifts are subnormal, Pacific J. Math.17 (1966), 367–379.
Wolfram Research, Inc.,Mathematica, Version 3.0, Wolfram Research, Inc., Champaign, IL, 1996.
D. Xia,Analytic theory of subnormal operators, Integral Equations Operator Theory10 (1987), 880–903.
D. Xia,On pure subnormal operators with finite rank self-commutators and related operator tuples, Integral Equations Operator Theory24 (1996), 107–125.
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Supported by NSF research grant DMS-9800931.
Supported by the Brain Korea 21 Project from the Korean Ministry of Education.
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Curto, R.E., Lee, W.Y. Towards a model theory for 2-hyponormal operators. Integr equ oper theory 44, 290–315 (2002). https://doi.org/10.1007/BF01212035
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DOI: https://doi.org/10.1007/BF01212035