Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

SUBNORMALITY OF S2(a, b, c, d) AND ITS BERGER MEASURE

  • Duan, Yongjiang (School of Mathematics and Statistics Northeast Normal University) ;
  • Ni, Jiaqi (School of Mathematics and Statistics Northeast Normal University)
  • Received : 2015.06.21
  • Published : 2016.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

We introduce a 2-variable weighted shift, denoted by $S_2$(a, b, c, d), which arises naturally from analytic function space theory. We investigate when it is subnormal, and compute the Berger measure of it when it is subnormal. And we apply the results to investigate the relationship among 2-variable subnormal, hyponormal and 2-hyponormal weighted shifts.

Keywords

References

  1. J. Conway, The Theory of Subnormal Operators, Mathematical Surveys and Monographs, vol. 36, Amer. Math. Soc., Providence, RI, 1991.
  2. R. Courant and F. John, Introduction to Calculus and Analysis Vol. 2, Springer-Verlag, New York, 1989.
  3. J. Cui and Y. Duan, Berger measure for S(a, b, c, d), J. Math. Anal. Appl. 413 (2014), no. 1, 202-211. https://doi.org/10.1016/j.jmaa.2013.11.053
  4. R. Curto, Joint hyponormality: A bridge between hyponormality and subnormality, Proc. Sympos. Pure Math. 51 (1990), Part 2, 69-91.
  5. R. Curto, S. Lee, and J. Yoon, k-Hyponormality of multivariable weighted shifts, J. Funct. Anal. 229 (2005), no. 2, 462-480. https://doi.org/10.1016/j.jfa.2005.03.022
  6. R. Curto, Y. Poon, and J. Yoon, Subnormality of Bergman-like weighted shifts, J. Math. Anal. Appl. 308 (2005), no. 1, 334-342. https://doi.org/10.1016/j.jmaa.2005.01.028
  7. R. Curto and J. Yoon, Spectral pictures of 2-variable weighted shifts, C. R. Math. Acad. Sci. Paris 343 (2006), no. 9, 579-584. https://doi.org/10.1016/j.crma.2006.09.024
  8. 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
  9. R. Curto and J. Yoon, Propagation phenomena for hyponormal 2-variable weighted shifts, J. Operator Theory 58 (2007), no. 1, 175-203.
  10. Y. Duan and T. Qi, Weakly k-hyponormal and polynomially hyponormal commuting operator pairs, Sci. China Math. 58 (2015), no. 2, 405-422. https://doi.org/10.1007/s11425-014-4916-x
  11. G. Exner, Aluthge transforms and n-contractivity of weighted shifts, J. Operator Theory 61 (2009), no. 2, 419-438.
  12. K. Guo, J. Hu, and X. Xu, Toeplitz algebras, subnormal tuples and rigidity on reproducing C[$z_{1}$,..., $z_{d}$]-modules, J. Funct. Anal. 210 (2004), no. 1, 214-247. https://doi.org/10.1016/j.jfa.2003.06.003
  13. N. Jewell and A. Lubin, Commuting weighted shifts and analytic function theory in several variables, J. Operator Theory 1 (1979), no. 2, 207-223.
  14. A. Shields, Weighted Shift Operator and Analytic Function Theory, Math Surveys, vol. 13. Amer. Math. Soc., Providence, 1974.
  15. K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Springer-Verlag, New York, 2005.