×

zbMATH — the first resource for mathematics

Commuting quasihomogeneous Toeplitz operators on the harmonic Bergman space. (English) Zbl 1291.47024
The authors consider Toeplitz operators with quasihomogeneous symbols acting on the harmonic Bergman space on the unit disk. The main results of the paper state that two such Toeplitz operators with generic quasihomogeneous symbols do not commute; their commutativity is possible only under some specific conditions on symbols trivializing the situation.

MSC:
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Axler S., Čučković Ž.: Commuting Toeplitz operators with harmonic symbols. Integr. Equ. Oper. Theory. 14, 1–12 (1991) · Zbl 0733.47027
[2] Brown A., Halmos P.R.: Algebraic properties of Toeplitz operators. J. Reine Angew. Math. 213, 89–102 (1964) · Zbl 0116.32501
[3] Choe B.R., Lee Y.J.: Commuting Toeplitz operators on the harmonic Bergman spaces. Michigan Math. J. 46, 163–174 (1999) · Zbl 0969.47023
[4] Choe B.R., Lee Y.J.: Commutants of analytic Toeplitz operators on the harmonic Bergman space. Integr. Equ. Oper. Theory. 50, 559–564 (2004) · Zbl 1088.47016
[5] Čučković Ž.: Commutants of Toeplitz operators on the Bergman space. Pacific J. Math. 162, 277–285 (1994) · Zbl 0802.47018
[6] Čučković Ž., Rao N.V.: Mellin transform, monomial symbols, and commuting Toeplitz operators. J. Funct. Anal. 154, 195–214 (1998) · Zbl 0936.47015
[7] Ding X.H.: A question of Toeplitz operators on the harmonic Bergman space. J. Math. Anal. Appl. 344, 367–372 (2008) · Zbl 1143.47018
[8] Dong X.T., Zhou Z.H.: Algebraic properties of Toeplitz operators with separately quasihomogeneous symbols on the Bergman space of the unit ball. J. Oper. Theory. 66, 193–207 (2011) · Zbl 1234.47014
[9] Dong X.T., Zhou Z.H.: Products of Toeplitz operators on the harmonic Bergman space. Proc. Amer. Math. Soc. 138, 1765–1773 (2010) · Zbl 1195.47014
[10] Louhichi I.: Powers and roots of Toeplitz operators. Proc. Amer. Math. Soc. 135, 1465–1475 (2007) · Zbl 1112.47023
[11] Louhichi I., Strouse E., Zakariasy L.: Products of Toeplitz operators on the Bergman space. Integr. Equ. Oper. Theory 54, 525–539 (2006) · Zbl 1109.47023
[12] Louhichi I., Zakariasy L.: On Toeplitz operators with quasihomogeneous symbols. Arch. Math. 85, 248–257 (2005) · Zbl 1088.47019
[13] Ohno S.: Toeplitz and Hankel operators on the harmonic Bergman spaces. RIMS Kokyuroku. 946, 25–34 (1996)
[14] Remmert R.: Classical Topics in Complex Function Theory. Graduate Texts in Methematics, Springer, New York (1998) · Zbl 0895.30001
[15] Zakariasy L.: The rank of Hankel operators on harmonic Bergman spaces. Proc. Amer. Math. Soc. 131, 1177–1180 (2003) · Zbl 1010.47018
[16] Zhou Z.H., Dong X.T.: Algebraic properties of Toeplitz operators with radial symbols on the Bergman space of the unit ball. Integr. Equ. Oper. Theory. 64, 137–154 (2009) · Zbl 1195.47020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.