Mean Oscillation and Hankel Operators on the Segal-Bargmann Space

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).