Essential Norm of the Weighted Composition Operators Between Growth Space - Ebrahim Abbasi; Mostafa Hassanlou (Sahand Communications in Mathematical Analysis)
For $α>0$, the growth space $\mathcal{A}^{-α}$ is the space of all function $f\in H(\DD)$ such that $$\left\|f\right\|_{\mathcal{A}^{-α}}=\sup_{z\in\DD}\left(1-\left|z\right|^2\right)^α \left|f(z)\right|<∞.$$In this work, we obtain exact formula for the norm of weighted composition operators from $\mathcal{A}^{-α}$ into $\mathcal{A}^{-β}$. Especially, we show that\begin{align*}\left\|uC_φ\right\|_{ \mathcal{A}^{-α}→ \mathcal{A}^{-β}}= \sup_{z\in\mathbb{D}}\frac{\left(1-\left |z \right|^2\right)^β \left|u(z)\right|}{\left(1-\left |φ(z) \right|^2 \right)^α}.\end{align*}As a corollary, we show that $C_φ: \mathcal{A}^{-α}→ \mathcal{A}^{-α}$ is isometry if and only if $\f$ is rotation. Then the exact formula for the essential norm $uC_φ: \mathcal{A}^{-α}→ \mathcal{A}^{-β}$ is given as follow$$\left\|uC_φ\right\|_{e, \mathcal{A}^{-α}→ \mathcal{A}^{-β}}= \left(\frac{e}{2α}\right)^α \limsup n^α \left\|uφ^{n-1}\right\|_{\mathcal{A}^{-β}}.$$Also, some equivalence conditions for compactness of such operators operator between difference growth spaces are given.