Berezin Transform, Mellin Transform and Toeplitz Operators

Abstract

We consider functions of the form ${f_1\bar g_1+h}$ in the range of the Berezin transform B, where f ₁ and g ₁ are holomorphic on the unit disk ${\mathbb D}$, and h is either harmonic or of the form ${f_2\bar g_2}$ for some holomorphic functions f ₂ and g ₂ on ${\mathbb D}$. First, by using the Mellin transform, we complement Ahern’s Theorem (Ahern in J Funct Anal 215:206–216, 2004) by proving that if ${u\in L^1}$ and B(u) is harmonic, then u is harmonic. Secondly, we extend Ahern’s Theorem when h is harmonic, and give very precise relations between f ₁ and f ₂, g ₁ and g ₂ when ${h=f_2\bar g_2}$ and g ₂(z) = z ⁿ with n ≥ 1. Finally, some applications of our results to the theory of Toeplitz operators are discussed.