Positivity of Fock Toeplitz Operators via the Berezin Transform

This paper deals with the relationship between the positivity of theFock Toeplitz operators and their Berezin transforms. The authorconsiders the special case of the bounded radial function$varphi(z)=a+brme^{-alpha|z|^2}+crme^{-beta|z|^2}$, where $a,b, c$ are real numbers and $alpha, beta$ are positive numbers. Forthis type of $varphi$, one can choose these parameters such thatthe Berezin transform of $varphi$ is a nonnegative function on thecomplex plane, but the corresponding Toeplitz operator $T_varphi$is not positive on the Fock space.