Commuting Toeplitz operators with pluriharmonic symbols on the Fock space. (English) Zbl 1327.47023
Authors’ abstract: In the setting of the Bergman space over the disk or the ball, it has been known that two Toeplitz operators with bounded pluriharmonic symbols can (semi-)commute only in the trivial cases. In this paper, we study the analogues on the Fock space over the multi-dimensional complex space. As is the case in various other settings, we are naturally led to the problem of characterizing a certain type of fixed points of the Berezin transform. For such fixed points, we obtain a complete characterization by means of eigenfunctions of the Laplacian. We also obtain other characterizations. In particular, it turns out that there are many nontrivial cases on the Fock space for (semi-)commuting Toeplitz operators with pluriharmonic symbols. All in all, our results reveal that the situation on the Fock space appears to be much more complicated than in the classical Bergman space setting, which partly is caused by the unboundedness of the operator symbols. Some of our results are restricted to the one-variable case and the corresponding several-variable case is left open.
Reviewer: Waleed Al-Rawashdeh (Butte)
MSC:
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 30H20 | Bergman spaces and Fock spaces |
| 32A37 | Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) |
References:
| [1] | Ahern, P., On the range of the Berezin transform, J. Funct. Anal., 215, 206-216 (2004) · Zbl 1088.47014 |
| [2] | Axler, S.; Čučuković, Z̆., Commuting Toeplitz operators with harmonic symbols, Integral Equations Operator Theory, 14, 1-11 (1991) · Zbl 0733.47027 |
| [3] | Bargmann, V., On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math., 14, 187-214 (1961) · Zbl 0107.09102 |
| [4] | Bauer, W., Berezin-Toeplitz quantization and composition formulas, J. Funct. Anal., 256, 3107-3142 (2009) · Zbl 1169.47017 |
| [5] | Bauer, W.; Issa, H., Commuting Toeplitz operators with quasi-homogeneous symbols on the Segal-Bargmann space, J. Math. Anal. Appl., 386, 1, 213-235 (2012) · Zbl 1228.47030 |
| [6] | Bauer, W.; Le, T., Algebraic properties and the finite rank problem for Toeplitz operators on the Segal-Bargmann space, J. Funct. Anal., 261, 9, 2617-2640 (2011) · Zbl 1339.47035 |
| [7] | Bauer, W.; Lee, Y. J., Commuting Toeplitz operators on the Segal-Bargmann space, J. Funct. Anal., 260, 2, 460-489 (2011) · Zbl 1307.47026 |
| [8] | Berger, C. A.; Coburn, L. A., Heat flow and Berezin-Toeplitz estimates, Amer. J. Math., 116, 563-590 (1994) · Zbl 0839.46018 |
| [9] | Brown, A.; Halmos, P., Algebraic properties of Toeplitz operators, J. Reine Angew. Math., 213, 89-102 (1964) · Zbl 0116.32501 |
| [10] | Choe, B. R.; Koo, H.; Lee, Y. J., Pluriharmonic symbols of commuting Toeplitz operators over the polydisk, Trans. Amer. Math. Soc., 256, 1727-1749 (2004) · Zbl 1060.47034 |
| [11] | Choe, B. R.; Koo, H.; Lee, Y. J., Sums of Toeplitz products with harmonic symbols, Rev. Mat. Iberoam., 24, 1, 43-70 (2008) · Zbl 1148.47024 |
| [12] | Choe, B. R.; Lee, Y. J., Pluriharmonic symbols of commuting Toeplitz operators, Illinois J. Math., 37, 424-436 (1993) · Zbl 0816.47024 |
| [13] | Čučuković, Z̆.; Rao, N. V., Mellin transform, monomial symbols and commuting Toeplitz operators, J. Funct. Anal., 154, 195-214 (1998) · Zbl 0936.47015 |
| [14] | Englis, M., Functions invariant under the Berezin transform, J. Funct. Anal., 121, 1, 233-254 (1994) · Zbl 0807.47017 |
| [15] | Englis, M., Berezin and Berezin-Toeplitz quantizations for general function spaces, Rev. Mat. Complut., 19, 2, 385-430 (2006) · Zbl 1122.53053 |
| [16] | Folland, G. B., Harmonic Analysis in Phase Space, Ann. of Math. Stud., vol. 122 (1989), Princeton Univ. Press · Zbl 0671.58036 |
| [17] | Grudsky, S.; Quiroga-Barranco, R.; Vasilevski, N., Commutative \(C^\ast \)-algebras of Toeplitz operators and quantization on the unit disc, J. Funct. Anal., 234, 1-44 (2006) · Zbl 1100.47023 |
| [18] | Krantz, S. G., Function Theory of Several Complex Variables (1992), Amer. Math. Soc.: Amer. Math. Soc. Providence · Zbl 0776.32001 |
| [19] | Le, T., The commutants of certain Toeplitz operators on weighted Bergman spaces, J. Math. Anal. Appl., 348, 1, 1-11 (2008) · Zbl 1168.47025 |
| [20] | Lee, Y. J., Commuting Toeplitz operators on the Hardy space of the polydisk, Proc. Amer. Math. Soc., 138, 1, 189-197 (2010) · Zbl 1189.47028 |
| [21] | Louhichi, I.; Rao, N. V., Bicommutants of Toeplitz operators, Arch. Math., 91, 256-264 (2008) · Zbl 1168.47026 |
| [23] | Schaefer, H. H., Topological Vector Spaces (1971), Springer: Springer New York · Zbl 0212.14001 |
| [24] | Vasilevski, N., Commutative Algebras of Toeplitz Operators on the Bergman Space, Oper. Theory Adv. Appl. (2008), Birkhäuser · Zbl 1168.47057 |
| [25] | Vasilevski, N., Quasi-radial quasi-homogeneous symbols and commutative Banach algebras of Toeplitz operators, Integral Equations Operator Theory, 66, 141-152 (2010) · Zbl 1216.47050 |
| [26] | Zheng, D., Semi-commutators of Toeplitz operators on the Bergman space, Integral Equations Operator Theory, 25, 347-372 (1996) · Zbl 0864.47015 |
| [27] | Zheng, D., Commuting Toeplitz operators with pluriharmonic symbols, Trans. Amer. Math. Soc., 350, 1595-1618 (1998) · Zbl 0893.47015 |
| [28] | Zhu, K., Operator Theory in Function Spaces, Mathematical Surveys and Monographs, vol. 138 (2007), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 1123.47001 |
| [29] | Zhu, K., Analysis on Fock Spaces (2012), Springer: Springer New York · Zbl 1262.30003 |
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