Schatten class Hankel operators on the Segal-Bargmann space and the Berger-Coburn phenomenon

Hu, Zhangjian; Virtanen, Jani

doi:10.1090/tran/8638

Schatten class Hankel operators on the Segal-Bargmann space and the Berger-Coburn phenomenon
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by Zhangjian Hu and Jani A. Virtanen PDF

Trans. Amer. Math. Soc. 375 (2022), 3733-3753 Request permission

Corrigendum: Trans. Amer. Math. Soc. (to appear).

Abstract:

We give a complete characterization of Schatten class Hankel operators $H_f$ acting on weighted Segal-Bargmann spaces $F^2(\varphi )$ using the notion of integral distance to analytic functions in $\mathbb {C}^n$ and Hörmander’s $\bar \partial$-theory. Using our characterization, for $f\in L^\infty$ and $1<p<\infty$, we prove that $H_f$ is in the Schatten class $S_p$ if and only if $H_{\bar {f}}\in S_p$, which was previously known only for the Hilbert-Schmidt class $S_2$ of the standard Segal-Bargmann space $F^2(\varphi )$ with $\varphi (z) = \alpha |z|^2$.

References

Lars V. Ahlfors, Lectures on quasiconformal mappings, 2nd ed., University Lecture Series, vol. 38, American Mathematical Society, Providence, RI, 2006. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. MR 2241787, DOI 10.1090/ulect/038
Kari Astala, Tadeusz Iwaniec, and Gaven Martin, Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton Mathematical Series, vol. 48, Princeton University Press, Princeton, NJ, 2009. MR 2472875
Wolfram Bauer, Hilbert-Schmidt Hankel operators on the Segal-Bargmann space, Proc. Amer. Math. Soc. 132 (2004), no. 10, 2989–2996. MR 2063120, DOI 10.1090/S0002-9939-04-07264-8
C. A. Berger and L. A. Coburn, Toeplitz operators on the Segal-Bargmann space, Trans. Amer. Math. Soc. 301 (1987), no. 2, 813–829. MR 882716, DOI 10.1090/S0002-9947-1987-0882716-4
So-Chin Chen and Mei-Chi Shaw, Partial differential equations in several complex variables, AMS/IP Studies in Advanced Mathematics, vol. 19, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001. MR 1800297, DOI 10.1090/amsip/019
Hong Rae Cho and Kehe Zhu, Fock-Sobolev spaces and their Carleson measures, J. Funct. Anal. 263 (2012), no. 8, 2483–2506. MR 2964691, DOI 10.1016/j.jfa.2012.08.003
Michael Christ, On the $\overline \partial$ equation in weighted $L^2$ norms in $\textbf {C}^1$, J. Geom. Anal. 1 (1991), no. 3, 193–230. MR 1120680, DOI 10.1007/BF02921303
Henrik Delin, Pointwise estimates for the weighted Bergman projection kernel in $\mathbf C^n$, using a weighted $L^2$ estimate for the $\overline \partial$ equation, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 4, 967–997 (English, with English and French summaries). MR 1656004, DOI 10.5802/aif.1645
Quanlei Fang and Jingbo Xia, Hankel operators on weighted Bergman spaces and norm ideals, Complex Anal. Oper. Theory 12 (2018), no. 3, 629–668. MR 3770363, DOI 10.1007/s11785-017-0710-4
Duane Farnsworth, Hankel operators, the Segal-Bargmann space, and symmetrically-normed ideals, J. Funct. Anal. 260 (2011), no. 5, 1523–1542. MR 2749437, DOI 10.1016/j.jfa.2010.10.022
R. Hagger and J. A. Virtanen, Compact Hankel operators with bounded symbols, J. Operator Theory 86 (2021), 317–329.
Zhangjian Hu and Xiaofen Lv, Toeplitz operators on Fock spaces $F^p(\varphi )$, Integral Equations Operator Theory 80 (2014), no. 1, 33–59. MR 3248473, DOI 10.1007/s00020-014-2168-3
Zhangjian Hu and Jani A. Virtanen, IDA and Hankel operators on Fock spaces, arXiv:2111.04821.
Zhangjian Hu and Ermin Wang, Hankel operators between Fock spaces, Integral Equations Operator Theory 90 (2018), no. 3, Paper No. 37, 20. MR 3803293, DOI 10.1007/s00020-018-2459-1
Lars Hörmander, $L^{2}$ estimates and existence theorems for the $\bar \partial$ operator, Acta Math. 113 (1965), 89–152. MR 179443, DOI 10.1007/BF02391775
J. Isralowitz, Schatten $p$ class Hankel operators on the Segal-Bargmann space $H^2(\Bbb C^n,\textrm {d}\mu )$ for $0<p<1$, J. Operator Theory 66 (2011), no. 1, 145–160. MR 2806550
Joshua Isralowitz, Jani Virtanen, and Lauren Wolf, Schatten class Toeplitz operators on generalized Fock spaces, J. Math. Anal. Appl. 421 (2015), no. 1, 329–337. MR 3250481, DOI 10.1016/j.jmaa.2014.05.065
Josh Isralowitz and Kehe Zhu, Toeplitz operators on the Fock space, Integral Equations Operator Theory 66 (2010), no. 4, 593–611. MR 2609242, DOI 10.1007/s00020-010-1768-9
Huiping Li, Hankel operators on the Bergman spaces of strongly pseudoconvex domains, Integral Equations Operator Theory 19 (1994), no. 4, 458–476. MR 1285493, DOI 10.1007/BF01299844
Huiping Li and Daniel H. Luecking, Schatten class of Hankel and Toeplitz operators on the Bergman space of strongly pseudoconvex domains, Multivariable operator theory (Seattle, WA, 1993) Contemp. Math., vol. 185, Amer. Math. Soc., Providence, RI, 1995, pp. 237–257. MR 1332063, DOI 10.1090/conm/185/02157
Daniel H. Luecking, Characterizations of certain classes of Hankel operators on the Bergman spaces of the unit disk, J. Funct. Anal. 110 (1992), no. 2, 247–271. MR 1194989, DOI 10.1016/0022-1236(92)90034-G
Daniel H. Luecking, Trace ideal criteria for Toeplitz operators, J. Funct. Anal. 73 (1987), no. 2, 345–368. MR 899655, DOI 10.1016/0022-1236(87)90072-3
Alexander P. Schuster and Dror Varolin, Toeplitz operators and Carleson measures on generalized Bargmann-Fock spaces, Integral Equations Operator Theory 72 (2012), no. 3, 363–392. MR 2891634, DOI 10.1007/s00020-011-1939-3
Karel Stroethoff, Hankel and Toeplitz operators on the Fock space, Michigan Math. J. 39 (1992), no. 1, 3–16. MR 1137884, DOI 10.1307/mmj/1029004449
Jingbo Xia and Dechao Zheng, Standard deviation and Schatten class Hankel operators on the Segal-Bargmann space, Indiana Univ. Math. J. 53 (2004), no. 5, 1381–1399. MR 2104282, DOI 10.1512/iumj.2004.53.2434
Kehe Zhu, Operator theory in function spaces, 2nd ed., Mathematical Surveys and Monographs, vol. 138, American Mathematical Society, Providence, RI, 2007. MR 2311536, DOI 10.1090/surv/138
Kehe Zhu, Analysis on Fock spaces, Graduate Texts in Mathematics, vol. 263, Springer, New York, 2012. MR 2934601, DOI 10.1007/978-1-4419-8801-0