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

Curto, Raúl; Yoon, Jasang

doi:10.1090/S0002-9947-06-03911-0

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.

[Ath] A. Athavale, On joint hyponormality of operators, Proc. Amer. Math. Soc. 103(1988), 417-423. MR 0943059 (89f:47033)
[Atk] Kendall E. Atkinson, An introduction to numerical analysis, 2nd ed., John Wiley & Sons, Inc., New York, 1989. MR 1007135
[Con] John B. Conway, The theory of subnormal operators, Mathematical Surveys and Monographs, vol. 36, American Mathematical Society, Providence, RI, 1991. MR 1112128
[Cu1] Raúl E. Curto, Joint hyponormality: a bridge between hyponormality and subnormality, Operator theory: operator algebras and applications, Part 2 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 69–91. MR 1077422
[Cu2] Raúl E. Curto, Quadratically hyponormal weighted shifts, Integral Equations Operator Theory 13 (1990), no. 1, 49–66. MR 1025673, https://doi.org/10.1007/BF01195292
[Cu3] Raúl E. Curto, An operator-theoretic approach to truncated moment problems, Linear operators (Warsaw, 1994) Banach Center Publ., vol. 38, Polish Acad. Sci. Inst. Math., Warsaw, 1997, pp. 75–104. MR 1457002
[CuFi1] Raúl E. Curto and Lawrence A. Fialkow, Recursiveness, positivity, and truncated moment problems, Houston J. Math. 17 (1991), no. 4, 603–635. MR 1147276
[CuFi2] Raúl E. Curto and Lawrence A. Fialkow, Recursively generated weighted shifts and the subnormal completion problem, Integral Equations Operator Theory 17 (1993), no. 2, 202–246. MR 1233668, https://doi.org/10.1007/BF01200218
[CuFi3] Raúl E. Curto and Lawrence A. Fialkow, Recursively generated weighted shifts and the subnormal completion problem. II, Integral Equations Operator Theory 18 (1994), no. 4, 369–426. MR 1265443, https://doi.org/10.1007/BF01200183
[CuFi4] Raúl E. Curto and Lawrence A. Fialkow, Solution of the truncated complex moment problem for flat data, Mem. Amer. Math. Soc. 119 (1996), no. 568, x+52. MR 1303090, https://doi.org/10.1090/memo/0568
[CuFi5] Raúl E. Curto and Lawrence A. Fialkow, Solution of the singular quartic moment problem, J. Operator Theory 48 (2002), no. 2, 315–354. MR 1938799
[CuLe] Raúl E. Curto and Woo Young Lee, Towards a model theory for 2-hyponormal operators, Integral Equations Operator Theory 44 (2002), no. 3, 290–315. MR 1933654, https://doi.org/10.1007/BF01212035
[CMX] Raúl E. Curto, Paul S. Muhly, and Jingbo Xia, Hyponormal pairs of commuting operators, Contributions to operator theory and its applications (Mesa, AZ, 1987) Oper. Theory Adv. Appl., vol. 35, Birkhäuser, Basel, 1988, pp. 1–22. MR 1017663
[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. Šmul′jan, An operator Hellinger integral, Mat. Sb. (N.S.) 49 (91) (1959), 381–430 (Russian). MR 0121662
[Sta] J. G. Stampfli, Which weighted shifts are subnormal?, Pacific J. Math. 17 (1966), 367–379. MR 0193520
[Wol] Wolfram Research, Inc. Mathematica, Version 4.2, Wolfram Research Inc., Champaign, IL, 2002.