Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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WHICH WEIGHTED SHIFTS ARE FLAT ?

  • SHEN, HAILONG (Department of Mathematics, Northeastern University) ;
  • LI, CHUNJI (Department of Mathematics, Northeastern University)
  • Received : 2020.04.08
  • Accepted : 2020.05.13
  • Published : 2020.09.30
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Abstract

The flatness property of a unilateral weighted shifts is important to study the gaps between subnormality and hyponormality. In this paper, we first summerize the results on the flatness for some special kinds of a weighted shifts. And then, we consider the flatness property for a local-cubically hyponormal weighted shifts, which was introduced in [2]. Let α : ${\sqrt{\frac{2}{3}}}$, ${\sqrt{\frac{2}{3}}}$, $\{{\sqrt{\frac{n+1}{n+2}}}\}^{\infty}_{n=2}$ and let Wα be the associated weighted shift. We prove that Wα is a local-cubically hyponormal weighted shift Wα of order ${\theta}={\frac{\pi}{4}}$ by numerical calculation.

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References

  1. S. Baek, G. Exner, I.B. Jung and C. Li, On semi-cubically hyponormal weighted shifts with first two equal weights, Kyungpook Math. J. 56 (2016), 899-910. https://doi.org/10.5666/KMJ.2016.56.3.899
  2. S. Baek, H. Do, M. Lee and C. Li, The flatness property of local-cubically hyponormal weighted shifts, Kyungpook Math. J. 59 (2019), 315-324. https://doi.org/10.5666/KMJ.2019.59.2.315
  3. Y.B. Choi, A propagation of quadratically hyponormal weighted shifts, Bull. Korean Math. Soc. 37 (2000) 347-352.
  4. R. Curto, Joint hyponormality: A bridge between hyponormality and subnormality, Proc. Sym. Math. 51 (1990), 69-91. https://doi.org/10.1090/pspum/051.2/1077422
  5. R. Curto, Quadratically hyponormal weighted shifts, Integr. Equ. Oper. Theory 13 (1990), 49-66. https://doi.org/10.1007/BF01195292
  6. R. Curto and L. Fialkow, Recursively generated weighted shifts and the subnormal completion problem, Integr. Equ. Oper. Theory 17 (1993), 202-246. https://doi.org/10.1007/BF01200218
  7. R. Curto and L. Fialkow, Recursively generated weighted shifts and the subnormal completion problem, II, Integral Equ. Oper. Theory 18 (1994), 369-426. https://doi.org/10.1007/BF01200183
  8. R. Curto and M. Putinar, Existence of non-subnormal polynomially hyponormal operators, Bull. Amer. Math. Soc. 25 (1991), 373-378. https://doi.org/10.1090/S0273-0979-1991-16079-9
  9. Y. Do, G. Exner, I.B. Jung and C. Li, On semi-weakly n-hyponormal weighted shifts, Integr. Equ. Oper. Theory 73 (2012), 93-106. https://doi.org/10.1007/s00020-012-1960-1
  10. G. Exner, I.B. Jung, and D.W. Park, Some quadratically hyponormal weighted shifts, Integr. Equ. Oper. Theory 60 (2008), 13-36. https://doi.org/10.1007/s00020-007-1544-7
  11. I.B. Jung and S.S. Park, Quadratically hyponormal weighted shifts and their examples, Integr. Equ. Oper. Theory 36 (2000), 480-498. https://doi.org/10.1007/BF01232741
  12. I.B. Jung and S.S. Park, Cubically hyponormal weighted shifts and their examples, J. Math. Anal. Appl. 247 (2000), 557-569. https://doi.org/10.1006/jmaa.2000.6879
  13. C. Li, A note on the local-cubic hyponormal weighted shifts, J. Appl. & Pure Math. 2 (2020), 1-7.
  14. C. Li, M. Cho and M.R. Lee, A note on cubically hyponormal weighted shifts, Bull. Korean Math. Soc. 51 (2014), 1031-1040. https://doi.org/10.4134/BKMS.2014.51.4.1031
  15. J. Stampfli, Which weighted shifts are subnormal, Pacific J. Math. 17 (1966), 367-379. https://doi.org/10.2140/pjm.1966.17.367
  16. MacKichan Software, Inc. Scientific WorkPlace, Version 4.0, MacKichan Software, Inc., 2002.