An estinatiion for the essential norm of Compositiion operators acting on Bloch-type spaces

TL;DR

This paper provides a new estimate for the essential norm of composition operators between Bloch-type spaces, linking it to the behavior of the composition with specific test functions near the boundary.

Contribution

It establishes a characterization of the essential norm of composition operators on Bloch-type spaces using boundary behavior of associated test functions.

Findings

Essential norm is comparable to a limsup involving boundary behavior.

Provides a new method to estimate the essential norm of composition operators.

Connects the essential norm to the behavior of special functions in Bloch spaces.

Abstract

Let \mu be any weight function defined on the unit disk $\Bbb D$ and let $\phi$ be an analytic self-map of $\Bbb D$. In the present paper we show that the essential norm of composition operator $C_\phi$ mapping from the $\alpha$-Bloch space, with $\alpha >0$, to $\mu$-Bloch space $\mathcal{B}^\mu$ is comparable to $$ \limsup_{|a|\to 1^-}\|\sigma_a\circ\phi\|_{\mathcal{B}^{\mu}}, $$ where, for $a\in\Bbb D$, $\sigma_a$ is a certain special function in $\alpha$-Bloch space.