Abstract.
For the Segal-Bargmann space of Gaussian square integrable entire functions on \( \mathbb{C}^{m} \) we consider Hankel operators H f with symbols in \( f \in \mathcal{T}(\mathbb{C}^{m} ). \) We completely characterize the functions in \( \mathcal{T}(\mathbb{C}^{m} ) \) for which the operators H f and \( H_{{\ifmmode\expandafter\bar\else\expandafter\=\fi{f}}} \) are simultaneously bounded or compact in terms of the mean oscillation of f. The analogous description holds for the commutators [M f , P] where M f denotes the “multiplication by f” and P is the Toeplitz projection. These results are already known in case of bounded symmetric domains Ω in \( \mathbb{C}^{m} \) (see [BBCZ] or [C]). In the present paper we combine some techniques of [BBCZ] and [BC1]. Finally, we characterize the entire function \( f \in \mathcal{H}(\mathbb{C}^{m} ) \cap \mathcal{T}(\mathbb{C}^{m} ) \) and the polynomials p in z and \( {\ifmmode\expandafter\bar\else\expandafter\=\fi{z}} \) for which the Hankel operators \( H_{{\ifmmode\expandafter\bar\else\expandafter\=\fi{f}}} \) and H p are bounded (resp. compact).
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Bauer, W. Mean Oscillation and Hankel Operators on the Segal-Bargmann Space. Integr. equ. oper. theory 52, 1–15 (2005). https://doi.org/10.1007/s00020-003-1272-6
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DOI: https://doi.org/10.1007/s00020-003-1272-6