Positivity of Toeplitz operators via Berezin transform

In this paper, we study positive Toeplitz operators on the Bergman space via their Berezin transforms. Surprisingly we show that the positivity of a Toeplitz operator on the Bergman space is not completely determined by the positivity of the Berezin transform of its symbol. In fact, we show that even if the minimal value of the Berezin transform of a quadratic polynomial of vertical bar z vertical bar on the unit disk is positive, the Toeplitz operator with the function as the symbol may not be positive. (C) 2014 Elsevier Inc. All rights reserved.