Essential Norm of the Weighted Composition Operators Between Growth Space

For $\alpha>0$, the growth space $\mathcal{A}^{-\alpha}$ is the space of all function $f\in H(\DD)$ such that $$\left\|f\right\|_{\mathcal{A}^{-\alpha}}=\sup_{z\in\DD}\left(1-\left|z\right|^2\right)^\alpha \left|f(z)\right|<\infty.$$In this work, we obtain exact formula for the norm  of weighted composition operators from $\mathcal{A}^{-\alpha}$ into $\mathcal{A}^{-\beta}$. Especially, we show that\begin{align*}\left\|uC_\varphi\right\|_{ \mathcal{A}^{-\alpha}\rightarrow \mathcal{A}^{-\beta}}= \sup_{z\in\mathbb{D}}\frac{\left(1-\left |z \right|^2\right)^\beta \left|u(z)\right|}{\left(1-\left |\varphi(z) \right|^2 \right)^\alpha}.\end{align*}As a corollary, we show that $C_\varphi: \mathcal{A}^{-\alpha}\rightarrow \mathcal{A}^{-\alpha}$ is isometry if and only if $\f$ is  rotation. Then the exact formula for the essential norm $uC_\varphi:  \mathcal{A}^{-\alpha}\rightarrow \mathcal{A}^{-\beta}$ is given as follow$$\left\|uC_\varphi\right\|_{e, \mathcal{A}^{-\alpha}\rightarrow \mathcal{A}^{-\beta}}= \left(\frac{e}{2\alpha}\right)^\alpha \limsup n^\alpha \left\|u\varphi^{n-1}\right\|_{\mathcal{A}^{-\beta}}.$$Also, some equivalence conditions for compactness of such operators operator  between difference growth spaces are given.