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Commuting Toeplitz operators on the Hardy space of the polydisk
Author:
Young Joo Lee
Journal:
Proc. Amer. Math. Soc. 138 (2010), 189-197
MSC (2000):
Primary 47B35; Secondary 32A35
Posted:
August 19, 2009
MathSciNet review:
2550183
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Abstract: We study the commuting problem for Toeplitz operators on the Hardy space of the polydisk. We consider two symbols where one is an arbitrarily bounded symbol and the other is a bounded symbol which has the form of a sum of an analytic function and a co-analytic function. We then give a necessary and sufficient condition on the symbols for which the corresponding Toeplitz operators commute each other.
- 1.
Sheldon
Axler and Željko
Čučković, Commuting Toeplitz operators with
harmonic symbols, Integral Equations Operator Theory
14 (1991), no. 1, 1–12. MR 1079815
(92f:47018), http://dx.doi.org/10.1007/BF01194925
- 2.
Arlen
Brown and P.
R. Halmos, Algebraic properties of Toeplitz operators, J.
Reine Angew. Math. 213 (1963/1964), 89–102. MR 0160136
(28 #3350)
- 3.
Boo
Rim Choe, Hyungwoon
Koo, and Young
Joo Lee, Commuting Toeplitz operators on the
polydisk, Trans. Amer. Math. Soc.
356 (2004), no. 5,
1727–1749 (electronic). MR 2031039
(2004k:47054), http://dx.doi.org/10.1090/S0002-9947-03-03430-5
- 4.
Boo
Rim Choe and Young
Joo Lee, Pluriharmonic symbols of commuting Toeplitz
operators, Illinois J. Math. 37 (1993), no. 3,
424–436. MR 1219648
(94i:47041)
- 5.
Xuanhao
Ding, Products of Toeplitz operators on the polydisk, Integral
Equations Operator Theory 45 (2003), no. 4,
389–403. MR 1971745
(2004b:47051), http://dx.doi.org/10.1007/s000200300013
- 6.
Xuanhao
Ding, The finite sum of finite products of Toeplitz operators on
the polydisk, J. Math. Anal. Appl. 320 (2006),
no. 1, 464–481. MR 2230483
(2007b:47072), http://dx.doi.org/10.1016/j.jmaa.2005.07.021
- 7.
Caixing
Gu, Some algebraic properties of Toeplitz and Hankel operators on
polydisk, Arch. Math. (Basel) 80 (2003), no. 4,
393–405. MR 1982839
(2004e:47039)
- 8.
Young
Joo Lee, Commuting Toeplitz operators on the Hardy space of the
bidisk, J. Math. Anal. Appl. 341 (2008), no. 1,
738–749. MR 2394121
(2009c:47037), http://dx.doi.org/10.1016/j.jmaa.2007.11.002
- 9.
Walter
Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New
York-Amsterdam, 1969. MR 0255841
(41 #501)
- 10.
Karel
Stroethoff, Algebraic properties of Toeplitz operators on the Hardy
space via the Berezin transform, Function spaces (Edwardsville, IL,
1998) Contemp. Math., vol. 232, Amer. Math. Soc., Providence, RI,
1999, pp. 313–319. MR 1678343
(2000a:47060), http://dx.doi.org/10.1090/conm/232/03407
- 11.
Dechao
Zheng, Commuting Toeplitz operators with
pluriharmonic symbols, Trans. Amer. Math.
Soc. 350 (1998), no. 4, 1595–1618. MR 1443898
(98i:47027), http://dx.doi.org/10.1090/S0002-9947-98-02051-0
- 1.
- S. Axler and Z. Cucković, Commuting Toeplitz operators with harmonic symbols, Integr. Equ. Oper. Theory 14(1991), 1-11. MR 1079815 (92f:47018)
- 2.
- A. Brown and P. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213(1964), 89-102. MR 0160136 (28:3350)
- 3.
- B. R. Choe, H. Koo and Y. J. Lee, Commuting Toeplitz operators on the polydisk, Trans. Amer. Math. Soc. 356(2004), 1727-1749. MR 2031039 (2004k:47054)
- 4.
- B. R. Choe and Y. J. Lee, Pluriharmonic symbols of commuting Toeplitz operators, Illinois J. Math. 37(1993), 424-436. MR 1219648 (94i:47041)
- 5.
- X. Ding, Products of Toeplitz operators on the polydisk, Integr. Equ. Oper. Theory 45(2003), 389-403. MR 1971745 (2004b:47051)
- 6.
- X. Ding, The finite sum of finite products of Toeplitz operators on the polydisk, J. Math. Anal. Appl. 320(2006), 464-481. MR 2230483 (2007b:47072)
- 7.
- C. Gu, Some algebraic properties of Toeplitz and Hankel operators on polydisk, Arch. Math. 80(2003), 393-405. MR 1982839 (2004e:47039)
- 8.
- Y. J. Lee, Commuting Toeplitz operators on the Hardy space of the bidisk, J. Math. Anal. Appl. 341(2008), 738-749. MR 2394121 (2009c:47037)
- 9.
- W. Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., 1969. MR 0255841 (41:501)
- 10.
- K. Stroethoff, Algebraic properties of Toeplitz operators on the Hardy space via the Berezin transform, Contemp. Math., 232, Amer. Math. Soc., Providence, RI, 1999, 313-319. MR 1678343 (2000a:47060)
- 11.
- D. Zheng, Commuting Toeplitz operators with pluriharmonic symbols, Trans. Amer. Math. Soc. 350(1998), 1595-1618. MR 1443898 (98i:47027)
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Additional Information
Young Joo Lee
Affiliation:
Department of Mathematics, Chonnam National University, Gwangju 500-757, Republic of Korea
Email:
leeyj@chonnam.ac.kr
DOI:
http://dx.doi.org/10.1090/S0002-9939-09-10073-4
PII:
S 0002-9939(09)10073-4
Keywords:
Toeplitz operators,
Hardy space,
polydisk.
Received by editor(s):
February 24, 2009
Posted:
August 19, 2009
Communicated by:
Marius Junge
Article copyright:
© Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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