Given a  BMO${}^1$  function f on $\mathbb{C}^n$,
  we can take the Berezin transform (better known as the ``heat
  transform") $\widetilde{f} ^{(t_0)}$ of $f$ with respect to the weighted
  Segal-Bargmann space  $H^2 (\mathbb{C}^n, d\mu_{t_0})$. In this talk,
  we will discuss the question of whether $\widetilde{f} ^{(t_0)}$
  vanishing at infinity for some $t_0 > 0$ implies that
  $\widetilde{f} ^{(t)}$ vanishes at infinity for  all $t > 0$, and we discuss
  the same question in the context of the weighted Bergman space of
  the unit ball.  Moreover, we discuss what implications these results
  have for the compactness of Toeplitz operators in both the weighted
  Bergman and Segal-Bargmann space situations, and finally briefly
  discuss recent work by H. Issa that extends of some of these results to
  the weighted Bergman space of general bounded symmetric domains.
  This is joint work with W. Bauer and L. Coburn.