Inequalities Involving Essential Norm Estimates of Product-Type Operators

Abstract

Consider an open unit disk $\mathbb{D}=\left\{z\in ℂ:\left|z\right|<1\right\}$ in the complex plane $ℂ$ , $\xi$ a holomorphic function on $\mathbb{D}$ , and $\psi$ a holomorphic self-map of $\mathbb{D}$ . For an analytic function $f$ , the weighted composition operator is denoted and defined as follows: $\left({\mathcal{W}}_{\xi ,\psi }f\right)\left(z\right)=\xi \left(z\right)f\left(\psi \left(z\right)\right)$ . We estimate the essential norm of this operator from Dirichlet-type spaces to Bers-type spaces and Bloch-type spaces.