Fredholm theory of Toeplitz operators on doubling Fock Hilbert spaces
DOI:
https://doi.org/10.7146/math.scand.a-120920Abstract
We study the Fredholm properties of Toeplitz operators acting on doubling Fock Hilbert spaces, and describe their essential spectra for bounded symbols of vanishing oscillation. We also compute the index of these Toeplitz operators in the special case when $\varphi (z) = \lvert {z}\rvert^{\beta }$ with $\beta >0$. Our work extends the recent results on Toeplitz operators on the standard weighted Fock spaces to the setting of doubling Fock spaces.
References
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Stein, E. M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993.
Stroethoff, K., Hankel and Toeplitz operators on the Fock space, Michigan Math. J. 39 (1992), no. 1, 3–16. https://doi.org/10.1307/mmj/1029004449
Zhu, K., Analysis on Fock spaces, Graduate Texts in Mathematics, 263. Springer, New York, 2012. https://doi.org/10.1007/978-1-4419-8801-0
Berger, C. A. and Coburn, L. A., Toeplitz operators on the Segal-Bargmann space, Trans. Amer. Math. Soc. 301 (1987), no. 2, 813–829. https://doi.org/10.2307/2000671
Cho, H. R. and Zhu, K., Fock-Sobolev spaces and their Carleson measures, J. Funct. Anal. 263 (2012), no. 8, 2483–2506. https://doi.org/10.1016/j.jfa.2012.08.003
Fulsche, R. and Hagger, R., Fredholmness of Toeplitz Operators on the Fock Space, Complex Anal. Oper. Theory 13 (2019), no. 2, 375–403. https://doi.org/10.1007/s11785-018-0803-8
Hu, Z. and Lv, X., Hankel operators on weighted Fock spaces, Sci. China Math. 46 (2016), no. 2, 141–156. https://doi.org/https://doi.org/10.1360/012015-19
Marco, N., Massaneda, X., and Ortega-Cerdà, J., Interpolating and sampling sequences for entire functions, Geom. Funct. Anal. 13 (2003), no. 4, 862–914. https://doi.org/10.1007/s00039-003-0434-7
Marzo, J. and Ortega-Cerdà, J., Pointwise estimates for the Bergman kernel of the weighted Fock space, J. Geom. Anal. 19 (2009), no. 4, 890–910. https://doi.org/10.1007/s12220-009-9083-x
Oliver, R. and Pascuas, D., Toeplitz operators on doubling Fock spaces, J. Math. Anal. Appl. 435 (2016), no. 2, 1426–1457. https://doi.org/10.1016/j.jmaa.2015.11.023
Schneider, G., Hankel operators with antiholomorphic symbols on the Fock space, Proc. Amer. Math. Soc. 132 (2004), no. 8, 2399–2409. https://doi.org/10.1090/S0002-9939-04-07362-9
Schneider, G. and Schneider, K. A., Generalized Hankel operators on the Fock space, Math. Nachr. 282 (2009), no. 12, 1811–1826. https://doi.org/10.1002/mana.200810169
Schuster, A. P. and Varolin, D., Toeplitz operators and Carleson measures on generalized Bargmann-Fock spaces, Integral Equations Operator Theory 72 (2012), no. 3, 363–392. https://doi.org/10.1007/s00020-011-1939-3
Stein, E. M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993.
Stroethoff, K., Hankel and Toeplitz operators on the Fock space, Michigan Math. J. 39 (1992), no. 1, 3–16. https://doi.org/10.1307/mmj/1029004449
Zhu, K., Analysis on Fock spaces, Graduate Texts in Mathematics, 263. Springer, New York, 2012. https://doi.org/10.1007/978-1-4419-8801-0
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Published
2020-09-03
How to Cite
Al-Qabani, A., Hilberdink, T., & Virtanen, J. A. (2020). Fredholm theory of Toeplitz operators on doubling Fock Hilbert spaces. MATHEMATICA SCANDINAVICA, 126(3), 593–602. https://doi.org/10.7146/math.scand.a-120920
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