Mean oscillation and Hankel operators on the Segal-Bargmann space

For the Segal-Bargmann space of Gaussian square integrable entire functions on C-m we consider Hankel operators H-f with symbols in f is an element of T(C-m). We completely characterize the functions in T(C-m) for which the operators Hf and Hr 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 Q in 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 is an element of H(C-m) boolean AND T(C-m) and the polynomials p in z and (z) over bar for which the Hankel operators H ((f)) over bar and H-p are bounded (resp. compact).