Essential norm estimates for weighted composition operator on the logarithmic Bloch space

TL;DR

This paper provides estimates for the essential norm of weighted composition operators on the logarithmic Bloch space, linking it to properties of the symbol functions and extending results to related growth and Zygmund spaces.

Contribution

It introduces new essential norm estimates for weighted composition operators on the logarithmic Bloch space and related spaces, advancing understanding of their boundedness and compactness.

Findings

Estimated the essential norm of $W_{u, \, \varphi}$ on the logarithmic Bloch space.

Derived bounds for the essential norm of composition operators on Logarithmic-Zygmund space.

Connected operator norms to properties of the symbol functions and space parameters.

Abstract

In this article, we estimate the essential norm of weighted composition operator $W_{u, \varphi}$, acting on the logarithmic Bloch space $\mathcal{B}^{v_{\log}}$, in terms of the $n$-power of the analytic function $\varphi$ and the norm of the $n$-power of the identity function. Also, we estimate the essential norm of the weighted composition operator from $\mathcal{B}^{v_{\log}}$ into the growth space $H_{v_{\log}}^{\infty}$. As a consequence of our result, we estimate the essential norm of the composition operator $C_\varphi$ acting on the Logarithmic-Zygmund space.