Products of Toeplitz operators on the Fock space

Cho, Hong Rae; Park, Jong-Do; Zhu, Kehe

doi:10.1090/S0002-9939-2014-12110-1

Products of Toeplitz operators on the Fock space
HTML articles powered by AMS MathViewer

by Hong Rae Cho, Jong-Do Park and Kehe Zhu PDF

Proc. Amer. Math. Soc. 142 (2014), 2483-2489 Request permission

Abstract:

Let $f$ and $g$ be functions, not identically zero, in the Fock space $F^2_\alpha$ of $\mathbb {C}^n$. We show that the product $T_fT_{\overline g}$ of Toeplitz operators on $F^2_\alpha$ is bounded if and only if $f(z)=e^{q(z)}$ and $g(z)=ce^{-q(z)}$, where $c$ is a nonzero constant and $q$ is a linear polynomial.

References

H. R. Cho, B. R. Choe and H. Koo, Linear combinations of composition operators on the Fock-Sobolev spaces, preprint.
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
Svante Janson, Jaak Peetre, and Richard Rochberg, Hankel forms and the Fock space, Rev. Mat. Iberoamericana 3 (1987), no. 1, 61–138. MR 1008445, DOI 10.4171/RMI/46
Jong-Do Park, Bounded Toeplitz products on the Bergman space of the unit ball in ${\Bbb C}^n$, Integral Equations Operator Theory 54 (2006), no. 4, 571–584. MR 2222985, DOI 10.1007/s00020-005-1405-1
V. P. Havin and N. K. Nikolski (eds.), Linear and complex analysis. Problem book 3. Part I, Lecture Notes in Mathematics, vol. 1573, Springer-Verlag, Berlin, 1994. MR 1334345
Karel Stroethoff and Dechao Zheng, Products of Hankel and Toeplitz operators on the Bergman space, J. Funct. Anal. 169 (1999), no. 1, 289–313. MR 1726756, DOI 10.1006/jfan.1999.3489
Karel Stroethoff and Dechao Zheng, Bounded Toeplitz products on the Bergman space of the polydisk, J. Math. Anal. Appl. 278 (2003), no. 1, 125–135. MR 1963469, DOI 10.1016/S0022-247X(02)00578-4
Karel Stroethoff and Dechao Zheng, Bounded Toeplitz products on Bergman spaces of the unit ball, J. Math. Anal. Appl. 325 (2007), no. 1, 114–129. MR 2273032, DOI 10.1016/j.jmaa.2006.01.009
Dechao Zheng, The distribution function inequality and products of Toeplitz operators and Hankel operators, J. Funct. Anal. 138 (1996), no. 2, 477–501. MR 1395967, DOI 10.1006/jfan.1996.0073
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