# WEAK AND QUADRATIC HYPONORMALITY OF 2-VARIABLE WEIGHTED SHIFTS AND THEIR EXAMPLES

• Li, Chunji (Department of Mathematics Northeastern University)
• Published : 2017.03.31
• 122 12

#### Abstract

Recently, Curto, Lee and Yoon considered the properties (such as, hyponormality, subnormality, and flatness, etc.) for 2-variable weighted shifts and constructed several families of commuting pairs of subnormal operators such that each family can be used to answer a conjecture of Curto, Muhly and Xia negatively. In this paper, we consider the weak and quadratic hyponormality of 2-variable weighted shifts ($W_1,W_2$). In addition, we detect the weak and quadratic hyponormality with some interesting 2-variable weighted shifts.

#### References

1. R. Curto and L. Fialkow, Recursively generated weighted shifts and the subnormal com-pletion problem, Integral Equations Operator Theory 17 (1993), no. 2, 202-246. https://doi.org/10.1007/BF01200218
2. R. Curto and L. Fialkow, Recursively generated weighted shifts and the subnormal completion problem. II, Integral Equations Operator Theory 18 (1994), no. 4, 369-426. https://doi.org/10.1007/BF01200183
3. R. Curto, S. Lee, and J. Yoon, A new approach to the 2-variable subnormal completion problem, J. Math. Anal. Appl. 370 (2010), no. 1, 270-283. https://doi.org/10.1016/j.jmaa.2010.04.061
4. R. Curto, S. Lee, and J. Yoon, One step extensions of subnormal 2-variable weighted shifts, Integral Equations Operator Theory 78 (2014), no. 3, 415-426. https://doi.org/10.1007/s00020-013-2121-x
5. R. Curto, P. Muhly, and J. Xia, Hyponormal pairs of commuting operators, Contributions to operator theory and its applications (Mesa, AZ, 1987), 1-22, Oper. Theory Adv. Appl., 35, Birkhauser, Basel, 1988.
6. R. Curto and J. Yoon, Jointly hyponormal pairs of commuting subnormal operators need not be jointly subnormal, Trans. Amer. Math. Soc. 358 (2006), no. 11, 5139-5159. https://doi.org/10.1090/S0002-9947-06-03911-0
7. R. Curto and J. Yoon, Disintegration-of-measure techniques for commuting multivariable weighted shifts, Proc. Lond. Math. Soc. 92 (2006), no. 2, 381-402. https://doi.org/10.1112/S0024611505015601
8. G. Exner, I. B. Jung, and S. S. Park, Weakly n-hyponormal weighted shifts and their examples, Integral Equations Opertor Theory 54 (2006), no. 2, 215-233. https://doi.org/10.1007/s00020-004-1360-2
9. J. Kim and J. Yoon, Flat phenomena of 2-variable weighted shifts, Linear Algebra Appl. 486 (2015), 234-254. https://doi.org/10.1016/j.laa.2015.08.010
10. J. Kim and J. Yoon, Hyponormality for commuting pairs of operators, J. Math. Anal. Appl. 434 (2016), no. 2, 1077-1090. https://doi.org/10.1016/j.jmaa.2015.09.058
11. C. Li, Two variable subnormal completion problem, Hokkaido Math. J. 32 (2003), no. 1, 21-29. https://doi.org/10.14492/hokmj/1350652422
12. C. Li and J. Wu, The characteristics of expansivity of two variable weighted shift, Integral Equations Operator Theory 65 (2009), no. 3, 405-414. https://doi.org/10.1007/s00020-009-1725-7
13. J. Yoon, Disintegration of measures and contractive 2-variable weighted shifts, Integral Equations Operator Theory 59 (2007), no. 2, 281-298. https://doi.org/10.1007/s00020-007-1509-x