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 Free Access

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]**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.

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