Essential Norm of the Weighted Composition Operators Between Growth Space

For \alpha>۰, 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(۱-\left|z\right|^۲\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(۱-\left |z \right|^۲\right)^\beta \left|u(z)\right|}{\left(۱-\left |\varphi(z) \right|^۲ \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}{۲\alpha}\right)^\alpha \limsup n^\alpha \left\|u\varphi^{n-۱}\right\|_{\mathcal{A}^{-\beta}}.Also, some equivalence conditions for compactness of such operators operator  between difference growth spaces are given.

کلیدواژه ها:

Essential norm ، Growth space ، Isometry ، norm