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Fredholm theory of Toeplitz operators on standard weighted Fock spaces. (English) Zbl 06964594

Summary: We study the Fredholm properties of Toeplitz operators with bounded symbols of vanishing mean oscillation in the complex plane. In particular, we prove that the Toeplitz operator with such a symbol is Fredholm on a standard weighted Fock space if and only if the Berezin transform of the symbol is bounded away from zero outside a sufficiently large disk in the complex plane. We also show that the Fredholm index of the Toeplitz operator can be computed via the winding of the symbol along a sufficiently large circle. We finish by considering Toeplitz operators with matrix-valued symbols.

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
30H20 Bergman spaces and Fock spaces
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[1] Bauer, W.: Mean oscillation and Hankel operators on the Segal–Bargmann space. - Integral Equations Operator Theory 52:1, 2005, 1–15. · Zbl 1067.47028
[2] Bauer, W., and J. Isralowitz: Compactness characterization of operators in the Toeplitz algebra of the Fock space Fαp. - J. Funct. Anal. 263:5, 2012, 1323–1355. · Zbl 1258.47044
[3] Berger, C. A., and L. A. Coburn: Toeplitz operators on the Segal–Bargmann space. - Trans. Amer. Math. Soc. 301:2, 1987, 813–829. · Zbl 0625.47019
[4] Berger, C. A., L. A. Coburn, and K. H. Zhu: Toeplitz operators and function theory in n-dimensions. - In: Pseudodifferential operators (Oberwolfach, 1986), Lecture Notes in Math. 1256, Springer, Berlin, 1987, 28–35.
[5] Böttcher, A., and B. Silbermann: Analysis of Toeplitz operators. - Springer Monogr. Math., Springer-Verlag, Berlin, second edition, 2006.
[6] Coburn, L. A., J. Isralowitz, and B. Li: Toeplitz operators with BMO symbols on the Segal–Bargmann space. - Trans. Amer. Math. Soc. 363:6, 2011, 3015–3030. · Zbl 1218.47044
[7] Fulsche, R., and R. Hagger: Fredholmness of Toeplitz operators on the Fock space. Complex Anal. Oper. Theory (to appear).
[8] Gohberg, I., and S. Goldberg, and M. A. Kaashoek: Classes of linear operators. Vol. I. - Oper. Theory Adv. Appl. 49, Birkhäuser Verlag, Basel, 1990.
[9] Hu, Z., and X. Lv: Toeplitz operators on Fock spaces Fp(ϕ). - Integral Equations Operator Theory 80:1, 2014, 33–59. · Zbl 1310.47048
[10] Hu, Z., and X. Lv: Hankel operators on weighted Fock spaces. - Sci. China Math. 46:2, 2016, 141–156 (in Chinese).
[11] Isralowitz, J., J. Virtanen, and L. Wolf: Schatten class Toeplitz operators on generalized Fock spaces. - J. Math. Anal. Appl. 421:1, 2015, 329–337. · Zbl 1306.47039
[12] Isralowitz, J., and K. Zhu: Toeplitz operators on the Fock space. - Integral Equations Operator Theory 66:4, 2010, 593–611. · Zbl 1218.47046
[13] Perälä, A., A. Schuster, and J. A. Virtanen: Hankel operators on Fock spaces. - In: Concrete operators, spectral theory, operators in harmonic analysis and approximation, Oper. Theory Adv. Appl. 236, Birkhäuser/Springer, Basel, 2014, 377–390.
[14] Perälä, A., J. Taskinen, and J. Virtanen: Toeplitz operators with distributional symbols on Fock spaces. - Funct. Approx. Comment. Math. 44:2, 2011, 203–213. Fredholm theory of Toeplitz operators on standard weighted Fock spaces783 · Zbl 1259.47037
[15] Schuster, A. P., and D. Varolin: Toeplitz operators and Carleson measures on generalized Bargmann–Fock spaces. - Integral Equations Operator Theory 72:3, 2012, 363–392. · Zbl 1262.47047
[16] Seip, K., and E. H. Youssfi: Hankel operators on Fock spaces and related Bergman kernel estimates. - J. Geom. Anal. 23:1, 2013, 170–201. · Zbl 1275.47063
[17] Stroethoff, K.: Hankel and Toeplitz operators on the Fock space. - Michigan Math. J. 39:1, 1992, 3–16. · Zbl 0774.47012
[18] Vasilevski, N. L.: Commutative algebras of Toeplitz operators on the Bergman space. - Oper. Theory Adv. Appl. 185, Birkhäuser Verlag, Basel, 2008. · Zbl 1168.47057
[19] Zhu, K.: Analysis on Fock spaces. - Graduate Texts in Math. 263, Springer, New York, 2012. Received 6 December 2017
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