Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

$k$-hyponormality of finite rank perturbations of unilateral weighted shifts


Authors: Raúl E. Curto and Woo Young Lee
Journal: Trans. Amer. Math. Soc. 357 (2005), 4719-4737
MSC (2000): Primary 47B20, 47B35, 47B37; Secondary 47-04, 47A20, 47A57
DOI: https://doi.org/10.1090/S0002-9947-05-04029-8
Published electronically: June 29, 2005
MathSciNet review: 2165385
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we explore finite rank perturbations of unilateral weighted shifts $W_{\alpha }$. First, we prove that the subnormality of $W_{\alpha }$ is never stable under nonzero finite rank perturbations unless the perturbation occurs at the zeroth weight. Second, we establish that 2-hyponormality implies positive quadratic hyponormality, in the sense that the Maclaurin coefficients of $D_{n}(s):=\text{det}\,P_{n}\,[(W_{\alpha }+sW_{\alpha }^{2})^{*},\, W_{\alpha }+s W_{\alpha }^{2}]\,P_{n}$are nonnegative, for every $n\ge 0$, where $P_{n}$ denotes the orthogonal projection onto the basis vectors $\{e_{0},\cdots ,e_{n}\}$. Finally, for $\alpha $ strictly increasing and $W_{\alpha }$ 2-hyponormal, we show that for a small finite-rank perturbation $\alpha ^{\prime }$ of $\alpha $, the shift $W_{\alpha ^{\prime }}$ remains quadratically hyponormal.


References [Enhancements On Off] (What's this?)

  • 1. A. Athavale, On joint hyponormality of operators, Proc. Amer. Math. Soc. 103 (1988), 417-423. MR 0943059 (89f:47033)
  • 2. J. Bram, Subnormal operators, Duke Math. J. 22 (1955), 75-94. MR 0068129 (16:835a)
  • 3. Y.B. Choi, J.K. Han and W.Y. Lee, One-step extension of the Bergman shift, Proc. Amer. Math. Soc. 128 (2000), 3639-3646. MR 1694855 (2001b:47037)
  • 4. J.B. Conway, The Theory of Subnormal Operators, Math. Surveys and Monographs, vol. 36, Amer. Math. Soc., Providence, 1991. MR 1112128 (92h:47026)
  • 5. J.B. Conway and W. Szymanski, Linear combination of hyponormal operators, Rocky Mountain J. Math. 18 (1988), 695-705. MR 0972659 (90a:47059)
  • 6. 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, Vol. 171, Longman, 1988, pp. 155-167. MR 0958573 (90j:47022)
  • 7. R.E. Curto, Quadratically hyponormal weighted shifts, Integral Equations Operator Theory 13 (1990), 49-66. MR 1025673 (90k:47061)
  • 8. -, Joint hyponormality: A bridge between hyponormality and subnormality, Proc. Sympos. Pure Math., vol. 51, Part 2, Amer. Math. Soc., Providence, 1990, pp. 69-91. MR 1077422 (91k:47049)
  • 9. -, An operator theoretic approach to truncated moment problems, in Linear Operators, Banach Center Publications 38 (1997), 75-104. MR 1457002 (99c:47014)
  • 10. R.E. Curto and L.A. Fialkow, Recursiveness, positivity, and truncated moment problems, Houston J. Math. 17 (1991), 603-635. MR 1147276 (93a:47016)
  • 11. -, Recursively generated weighted shifts and the subnormal completion problem, Integral Equations Operator Theory 17 (1993), 202-246. MR 1233668 (94h:47050)
  • 12. -, Recursively generated weighted shifts and the subnormal completion problem, II, Integral Equations Operator Theory 18 (1994), 369-426. MR 1265443 (94m:47044)
  • 13. 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. MR 1862138 (2002i:47041)
  • 14. R.E. Curto and W.Y. Lee, Joint hyponormality of Toeplitz pairs, Mem. Amer. Math. Soc. no. 712, Amer. Math. Soc., Providence, 2001. MR 1810770 (2002c:47042)
  • 15. -, Towards a model theory for $2$-hyponormal operators, Integral Equations Operator Theory 44 (2002), 290-315. MR 1933654 (2003m:47036)
  • 16. 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, pp. 1-22. MR 1017663 (90m:47037)
  • 17. R.E. Curto and M. Putinar, Existence of non-subnormal polynomially hyponormal operators, Bull. Amer. Math. Soc. (N.S.) 25 (1991), 373-378. MR 1091568 (93e:47028)
  • 18. -, Nearly subnormal operators and moment problems, J. Funct. Anal. 115 (1993), 480-497. MR 1234402 (95d:47024)
  • 19. R.G. Douglas, V.I. Paulsen, and K. Yan, Operator theory and algebraic geometry, Bull. Amer. Math. Soc. (N.S.) 20 (1989), 67-71. MR 0955316 (90f:47028)
  • 20. P.R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887-933. MR 0270173 (42:5066)
  • 21. -, A Hilbert Space Problem Book, $2$nd ed., Springer, New York, 1982. MR 0675952 (84e:47001)
  • 22. S. McCullough and V. Paulsen, A note on joint hyponormality, Proc. Amer. Math. Soc. 107 (1989), 187-195. MR 0972236 (90a:47062)
  • 23. A. Shields, Weighted shift operators and analytic function theory, Math. Surveys 13 (1974), 49-128. MR 0361899 (50:14341)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 47B20, 47B35, 47B37, 47-04, 47A20, 47A57

Retrieve articles in all journals with MSC (2000): 47B20, 47B35, 47B37, 47-04, 47A20, 47A57


Additional Information

Raúl E. Curto
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
Email: rcurto@math.uiowa.edu

Woo Young Lee
Affiliation: Department of Mathematics, Seoul National University, Seoul 151-742, Korea
Email: wylee@math.snu.ac.kr

DOI: https://doi.org/10.1090/S0002-9947-05-04029-8
Keywords: Weighted shifts, perturbations, subnormal, $k$-hyponormal, weakly $k$-hyponormal
Received by editor(s): December 10, 1999
Received by editor(s) in revised form: December 31, 2001
Published electronically: June 29, 2005
Additional Notes: The work of the first-named author was partially supported by NSF research grants DMS-9800931 and DMS-0099357.
The work of the second-named author was partially supported by a grant (R14-2003-006-01001-0) from the Korea Science and Engineering Foundation.
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society