Derivatives of the Berezin transform (Q416302)

Summary: For a rotation invariant domain \(\Omega\), we consider the Bergman space \(A^2(\Omega, \mu)\) and we investigate some properties of the rank one projection \(A(z) : = \langle \cdot, k_z\rangle k_z\). We prove that the trace of all the strong derivatives of \(A(z)\) is zero. We also focus on the generalized Fock space \(A^2(\mu_m)\), where \(\mu_m\) is the measure with weight \(e^{-|z|^m}, ~m > 0\), with respect to the Lebesgue measure on \(\mathbb C^n\) and establish estimations of derivatives of the Berezin transform of a bounded operator \(T\) on \(A^2(\mu_m)\).