A new criterion for -hyponormality via weak subnormality

Authors:
Raúl E. Curto, Sang Hoon Lee and Woo Young Lee

Journal:
Proc. Amer. Math. Soc. **133** (2005), 1805-1816

MSC (2000):
Primary 47B20, 47B35, 47A63; Secondary 47B37, 47B38, 47A05, 30D50

DOI:
https://doi.org/10.1090/S0002-9939-04-07727-5

Published electronically:
December 20, 2004

MathSciNet review:
2120281

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this article we obtain a criterion for -hyponormality via weak subnormality. Using this criterion we recapture Spitkovskii's subnormality criterion and give a simple proof of the main result in Gu's preprint (2001), which describes a gap between -hyponormality and ()-hyponormality for Toeplitz operators. In addition, we notice that the minimal normal extension of a subnormal operator is exactly the inductive limit of its minimal partially normal extensions.

**[Bra]**J. Bram,*Subnormal operators*, Duke Math. J.**22**(1955), 75-94. MR**0068129 (16:835a)****[Con]**J. B. Conway,*The Theory of Subnormal Operators*, Math. Surveys and Monographs vol. 36, Amer. Math. Soc., Providence, 1991. MR**1112128 (92h:47026)****[Cow1]**C. Cowen,*More subnormal Toeplitz operators*, J. Reine Angew. Math.**367**(1986), 215-219. MR**0839133 (87h:47063)****[Cow2]**-,*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)****[CoL]**C. C. Cowen and J. J. Long,*Some subnormal Toeplitz operators*, J. Reine Angew. Math.**351**(1984), 216-220. MR**0749683 (86h:47034)****[Cu1]**R.E. Curto,*Quadratically hyponormal weighted shifts*, Integral Equations Operator Theory**13**(1990), 49-66. MR**1025673 (90k:47061)****[Cu2]**-,*Joint hyponormality: A bridge between hyponormality and subnormality*, Operator Theory: Operator Algebras and Applications (Durham, NH, 1988) (W.B. Arveson and R.G. Douglas, eds.), Proc. Sympos. Pure Math., Vol.**51**, part II, American Mathematical Society, Providence, (1990), Part 11, 69-91. MR**1077422 (91k:47049)****[CuF1]**R.E. Curto and L.A. Fialkow,*Recursiveness, positivity, and truncated moment problems*, Houston J. Math.**17**(1991), 603-635. MR**1147276 (93a:47016)****[CuF2]**-,*Recursively generated weighted shifts and the subnormal completion problem*, Integral Equations Operator Theory**17**(1993), 202-2 46. MR**1233668 (94h:47050)****[CuF3]**-,*Recursively generated weighted shifts and the subnormal completion problem, II*, Integral Equations Operator Theory**18**(1994), 36 9-426. MR**1265443 (94m:47044)****[CJP]**R. E. Curto, I. B. Jung and S. S. Park,*A characterization of**-hyponormality via weak subnormality*, J. Math. Anal. Appl.**279**(2003), 556-568. MR**1974045 (2004b:47032)****[CLL]**R. E. Curto, S. H. Lee and W. Y. Lee,*Subnormality and 2-hyponormality for Toeplitz operators*, Integral Equations Operator Theory**44**(2002), 138- 148. MR**1930833 (2003f:47045)****[CuL1]**R. E. Curto and W. Y. Lee,*Joint hyponormality of Toeplitz pairs*, Memoirs Amer. Math. Soc. no. 712, Amer. Math. Soc., Providence, 2001. MR**1810770 (2002c:47042)****[CuL2]**-,*Towards a model theory for**-hyponormal operators*, Integral Equations Operator Theory**44**(2002), 290-315. MR**1933654 (2003m:47036)****[CuL3]**-,*Subnormality and**-hyponormality of Toeplitz operators: A brief survey and open questions*, Operator theory and Banach algebras (Rabat, 1999), 73-81, Theta, Bucharest, 2003. MR**2006315****[CMX]**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), 1-22. MR**1017663 (90m:47037)****[DPY]**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)****[Fan]**P. Fan,*Note on subnormal weighted shifts*, Proc. Amer. Math. Soc.**103**(1988), 801-802. MR**0947661 (89j:47016)****[Gu]**C. Gu,*Non-subnormal**-hyponormal Toeplitz operators*, preprint, 2001.**[Ha1]**P. R. Halmos,*Ten problems in Hilbert space*, Bull. Amer. Math. Soc.**76**(1970), 887-933. MR**0270173 (42:5066)****[Ha2]**-,*Ten years in Hilbert space*, Integral Equations Operator Theory**2**(1979), 529-564. MR**0555777 (81c:47003)****[MaZ]**J. Ma and S. Zhou,*A necessary and sufficient condition for an operator to be subnormal*, Nanjing Daxue Xuebao (Chinese)**2**(1985), 258-267. MR**0834313 (87i:47031)****[McCP]**S. McCullough and V. Paulsen,*A note on joint hyponormality*, Proc. Amer. Math. Soc.**107**(1989), 187-195. MR**0972236 (90a:47062)****[Smu]**J. L. Smul'jan,*An operator Hellinger integral (*Russian*)*, Mat. Sb. (N.S.)**91**(1959), 381-430. MR**0121662 (22:12396)****[Spi]**I. M. Spitkovskii,*A criterion for normality of operators in Hilbert space*, Funct. Anal. Appl.**16**(1982), 367-379. MR**0659177 (83i:47033)****[Sta]**J. G. Stampfli,*Which weighted shifts are subnormal*?, Pacific J. Math.**17**(1966), 367-379. MR**0193520 (33:1740)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
47B20,
47B35,
47A63,
47B37,
47B38,
47A05,
30D50

Retrieve articles in all journals with MSC (2000): 47B20, 47B35, 47A63, 47B37, 47B38, 47A05, 30D50

Additional Information

**Raúl E. Curto**

Affiliation:
Department of Mathematics, University of Iowa, Iowa City, Iowa 52242

Email:
rcurto@math.uiowa.edu

**Sang Hoon Lee**

Affiliation:
Department of Mathematics, Seoul National University, Seoul 151-742, Korea

Email:
shlee@math.skku.ac.kr

**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/S0002-9939-04-07727-5

Keywords:
$k$-hyponormal operators,
subnormal operators,
Toeplitz operators,
unilateral weighted shifts,
weak subnormality

Received by editor(s):
August 31, 2003

Received by editor(s) in revised form:
February 23, 2004

Published electronically:
December 20, 2004

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 third-named author was partially supported by KOSEF research project No. R01-2000-00003-0

Communicated by:
David R. Larson

Article copyright:
© Copyright 2004
American Mathematical Society