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Keywords:
variable exponent Bloch space; composition operator; difference; boundedness from below
variable exponent Bloch space; composition operator; difference; boundedness from below
Summary:
We consider the composition operator $C_{\varphi }$ on the variable exponent Bloch space $\mathcal {B}^{\alpha ({\cdot })}$, which consists of all analytic functions $f$ on the unit disk $\mathbb {D}$ such that $$ \sup \{(1-|z|^2)^{\alpha (z)}|f'(z)| \colon z\in \mathbb {D} \}<\infty . $$ Here, $\alpha (z)$ is a log-Hölder continuous function on $\mathbb {D}$. The boundedness and compactness of $C_{\varphi }$ are characterized. Besides, we show that $(1-|z|^2)^{\alpha (z)}f'(z)$ is Lipschitz continuous in terms of the pseudo-hyperbolic metric under the Lipschitz continuity of $\alpha (z)$. By using this result, we study the bounded and compact difference $C_{\varphi }-C_{\psi }$ of two composition operators on $\mathcal {B}^{\alpha ({\cdot })}$, and the boundedness from below of $C_{\varphi }$ is partially described.
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