Intertwining properties of bounded linear operators on the Bergman space

Abstract

In this paper we find conditions on $\phi, \psi\in L^{\infty}(\mathbb D)$ that are necessary and sufficient for the existence of bounded linear operators $S,T$ from the Bergman space $L_a^2(\mathbb D)$ into itself such that for all $z\in \mathbb D,$ $ \phi(z)=\langle Sk_z, k_z, \rangle, \psi(z)=\langle Tk_z, k_z \rangle$ and $C_aS=TC_a$ for all $a\in \mathbb D$ where $C_af=f\circ \phi_a$ for all $f\in L_a^2(\mathbb D)$ and $\phi_a(z)=\frac{a-z}{1-\bar a z}, z\in \mathbb D.$ Applications of the results are also discussed.