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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
Posted:
June 15, 2006
MathSciNet review:
2231888
Full-text PDF Free Access
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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.
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Ameer
Athavale, On joint hyponormality of
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103 (1988), no. 2,
417–423. MR
943059 (89f:47033), http://dx.doi.org/10.1090/S0002-9939-1988-0943059-X
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John
B. Conway, The theory of subnormal operators, Mathematical
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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
(91k:47049)
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Raúl
E. Curto, Quadratically hyponormal weighted shifts, Integral
Equations Operator Theory 13 (1990), no. 1,
49–66. MR
1025673 (90k:47061), http://dx.doi.org/10.1007/BF01195292
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Raúl
E. Curto, An operator-theoretic approach to truncated moment
problems, Linear operators (Warsaw, 1994) Banach Center Publ.,
vol. 38, Polish Acad. Sci., Warsaw, 1997, pp. 75–104. MR 1457002
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Raúl
E. Curto and Lawrence
A. Fialkow, Recursiveness, positivity, and truncated moment
problems, Houston J. Math. 17 (1991), no. 4,
603–635. MR 1147276
(93a:47016)
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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
(94h:47050), http://dx.doi.org/10.1007/BF01200218
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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
(94m:47044), http://dx.doi.org/10.1007/BF01200183
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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
(96g:47009), http://dx.doi.org/10.1090/memo/0568
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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 (2003j:47017)
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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
(2003m:47036), http://dx.doi.org/10.1007/BF01212035
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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
(90m:47037)
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M. Dritschel and S. McCullough, private communication.
- [JeLu]
Nicholas
P. Jewell and A.
R. Lubin, Commuting weighted shifts and analytic function theory in
several variables, J. Operator Theory 1 (1979),
no. 2, 207–223. MR 532875
(80c:47030)
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Ju.
L. Šmul′jan, An operator Hellinger integral, Mat.
Sb. (N.S.) 49 (91) (1959), 381–430 (Russian). MR 0121662
(22 #12396)
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J.
G. Stampfli, Which weighted shifts are subnormal?, Pacific J.
Math. 17 (1966), 367–379. MR 0193520
(33 #1740)
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Wolfram Research, Inc. Mathematica, Version 4.2, Wolfram Research Inc., Champaign, IL, 2002.
- [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
 -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.
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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:
http://dx.doi.org/10.1090/S0002-9947-06-03911-0
PII:
S 0002-9947(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
Posted:
June 15, 2006
Additional Notes:
This research was partially supported by NSF Grant DMS-0099357
Article copyright:
© Copyright 2006 American Mathematical Society
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