In this paper, it is shown that the Berezin-Toeplitz operator T g is compact or in the Schatten class S p of the Segal-Bargmann space for 1 p < ∞ wheneverg (s) ∈ C 0 (C n ) (vanishes at infinity) org (s) ∈ L p (C n , dv), respectively, for some s with 0 < s < 1 4 , whereg (s) is the heat transform of g on C n . Moreover, we show that compactness of T g implies thatg (s) is in C 0 (C n ) for all s > 1 4 and use this to show that, for g ∈ BMO 1 (C n ), we haveg (s) is in C 0 (C n ) for some s > 0 only ifg (s) is in C 0 (C n ) for all s > 0. This "backwards heat flow" result seems to be unknown for g ∈ BMO 1 and even g ∈ L ∞ . Finally, we show that our compactness and vanishing "backwards heat flow" results hold in the context of the weighted Bergman space L 2 a (B n , dv α ), where the "heat flow"g (s) is replaced by the Berezin transform B α (g) on L 2 a (B n , dv α ) for α > −1.
Consider two Toeplitz operators T g , T f on the Segal-Bargmann space over the complex plane. Let us assume that g is a radial function and both operators commute. Under certain growth condition at infinity of f and g we show that f must be radial, as well. We give a counterexample of this fact in case of bounded Toeplitz operators but a fast growing radial symbol g. In this case the vanishing commutator [T g , T f ] = 0 does not imply the radial dependence of f . Finally, we consider Toeplitz operators on the Segal-Bargmann space over C n and n > 1, where the commuting property of Toeplitz operators can be realized more easily.
For the Segal-Bargmann space of Gaussian square integrable entire functions on C m we consider Hankel operators H f with symbols in f ∈ T (C m ). We completely characterize the functions in T (C m ) for which the operators H f and Hf 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 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 ∈ H(C m ) ∩ T (C m ) and the polynomials p in z andz for which the Hankel operators Hf and Hp are bounded (resp. compact).
Mathematics Subject Classification (2000). Primary 47B10, 47B32, 47B35.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.