Toeplitz operators between Bergman–Orlicz spaces

Abstract

Given a positive Borel measure $\mu$ on the unit disk ${\mathbb {D}}$, let $K^\alpha _z(w)=\frac{1}{(1-\overline{z}w)^{2+\alpha }}$ be the reproducing kernel of $A_\alpha ^2({\mathbb {D}})$ at z. The Toeplitz operators with symbol $\mu$ are densely defined as follows:

$$\begin{aligned} T_\mu (f)(z)= \int _{{\mathbb {D}}}f(w)\overline{K^\alpha _z(w)}{\text {d}}\mu (w),~f\in H^\infty ({\mathbb {D}}). \end{aligned}$$

Using the tools such as Carleson measures, Berezin transform and the average functions, we characterize the boundedness and compactness of Toeplitz operators $T_\mu$ acting between two different Bergman–Orlicz spaces $A_\alpha ^{\Phi _1}({\mathbb {D}})$ and $A_\alpha ^{\Phi _2}({\mathbb {D}})$ for two convex growth functions $\Phi _1$ and $\Phi _2$.