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)

 
 

 

Jointly hyponormal pairs of commuting subnormal operators need not be jointly subnormal


Authors: Raúl E. Curto and Jasang Yoon
Journal: Trans. Amer. Math. Soc. 358 (2006), 5139-5159
MSC (2000): Primary 47B20, 47B37, 47A13, 28A50; Secondary 44A60, 47-04, 47A20
DOI: https://doi.org/10.1090/S0002-9947-06-03911-0
Published electronically: June 15, 2006
MathSciNet review: 2231888
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We construct three different families of commuting pairs of subnormal operators, jointly hyponormal but not admitting commuting normal extensions. Each such family can be used to answer in the negative a 1988 conjecture of R. Curto, P. Muhly and J. Xia. We also obtain a sufficient condition under which joint hyponormality does imply joint subnormality.


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

  • [Ath] A. Athavale, On joint hyponormality of operators, Proc. Amer. Math. Soc. 103(1988), 417-423. MR 0943059 (89f:47033)
  • [Atk] K. Atkinson, Introduction to Numerical Analysis, Wiley and Sons, 2nd. Ed. 1989. MR 1007135 (90m:65001)
  • [Con] J. Conway, The Theory of Subnormal Operators, Mathematical Surveys and Monographs, vol. 36, Amer. Math. Soc., Providence, 1991. MR 1112128 (92h:47026)
  • [Cu1] R. Curto, Joint hyponormality: A bridge between hyponormality and subnormality, Proc. Symposia Pure Math. 51(1990), 69-91. MR 1077422 (91k:47049)
  • [Cu2] R. Curto, Quadratically hyponormal weighted shifts, Integral Equations Operator Theory 13(1990), 49-66. MR 1025673 (90k:47061)
  • [Cu3] R. Curto, An operator-theoretic approach to truncated moment problems, in Linear Operators, Banach Center Publ., vol. 38, 1997, pp. 75-104. MR 1457002 (99c:47014)
  • [CuFi1] R. Curto and L. Fialkow, Recursiveness, positivity, and truncated moment problems, Houston J. Math. 17(1991), 603-635. MR 1147276 (93a:47016)
  • [CuFi2] R. Curto and L. Fialkow, Recursively generated weighted shifts and the subnormal completion problem, Integral Equations Operator Theory 17(1993), 202-246. MR 1233668 (94h:47050)
  • [CuFi3] R. Curto and L. Fialkow, Recursively generated weighted shifts and the subnormal completion problem, II, Integral Equations Operator Theory 18(1994), 369-426. MR 1265443 (94m:47044)
  • [CuFi4] R. Curto and L. Fialkow, Solution of the truncated complex moment problem with flat data, Memoirs Amer. Math. Soc. vol. 119, no. 568, Amer. Math. Soc., Providence, 1996. MR 1303090 (96g:47009)
  • [CuFi5] R. Curto and L. Fialkow, Solution of the singular quartic moment problem, J. Operator Theory 48(2002), 315-354. MR 1938799 (2003j:47017)
  • [CuLe] R. Curto and W.Y. Lee, Towards a model theory for $ 2$ -hyponormal operators, Integral Equations Operator Theory 44(2002), 290-315. MR 1933654 (2003m:47036)
  • [CMX] R. Curto, P. Muhly and J. Xia, Hyponormal pairs of commuting operators, Operator Theory: Adv. Appl. 35(1988), 1-22. MR 1017663 (90m:47037)
  • [DrMcC] M. Dritschel and S. McCullough, private communication.
  • [JeLu] N.P. Jewell and A.R. Lubin, Commuting weighted shifts and analytic function theory in several variables, J. Operator Theory 1(1979), 207-223. MR 0532875 (80c:47030)
  • [Smu] Ju. L. Smul'jan, An operator Hellinger integral, Mat. Sb. (N.S.) 49 (1959), 381-430 (in Russian). MR 0121662 (22:12396)
  • [Sta] J. Stampfli, Which weighted shifts are subnormal?, Pacific J. Math. 17(1966), 367-379. MR 0193520 (33:1740)
  • [Wol] Wolfram Research, Inc. Mathematica, Version 4.2, Wolfram Research Inc., Champaign, IL, 2002.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 47B20, 47B37, 47A13, 28A50, 44A60, 47-04, 47A20

Retrieve articles in all journals with MSC (2000): 47B20, 47B37, 47A13, 28A50, 44A60, 47-04, 47A20


Additional Information

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

Jasang Yoon
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
Email: jyoon@iastate.edu

DOI: https://doi.org/10.1090/S0002-9947-06-03911-0
Keywords: Jointly hyponormal pairs, subnormal pairs, $2$-variable weighted shifts
Received by editor(s): January 22, 2004
Received by editor(s) in revised form: December 5, 2004
Published electronically: June 15, 2006
Additional Notes: This research was partially supported by NSF Grant DMS-0099357
Article copyright: © Copyright 2006 American Mathematical Society

American Mathematical Society