Positive Toeplitz operators on Large Bergman spaces in the unit disk

We study positive Toeplitz operators on the Bergman spaces having the fast decreasing weight functions in a certain class. We give the characterizations for the boundedness and compactness of Toeplitz operators in terms of their Berezin transforms and averaging functions. In particular, we show the equivalent conditions in order that Toeplitz operators on $A^2(\omega)$ belong to the Schatten class $S_p$, $0<p<\infty$.