Properties of $\beta$-Ces\`aro operators on $\alpha$-Bloch space

TL;DR

This paper studies the properties of the $eta$-Cesàro operator on $ ext{alpha}$-Bloch spaces, focusing on boundedness, compactness, spectrum, and essential norm, providing a comprehensive operator analysis.

Contribution

It introduces and analyzes the $eta$-Cesàro operator on $ ext{alpha}$-Bloch spaces, including boundedness, compactness, spectrum, and essential norm calculations.

Findings

The $eta$-Cesàro operator is bounded on $ ext{alpha}$-Bloch spaces under certain conditions.

Conditions for the compactness of the $eta$-Cesàro operator are established.

The spectrum and essential norm of these operators are explicitly computed.

Abstract

For each $ \alpha > 0 $, the $\alpha$-Bloch space is consisting of all analytic functions $f$ on the unit disk satisfying $ \sup_{|z|<1} (1-|z|^2)^\alpha |f'(z)| < + \infty.$ In this paper, we consider the following complex integral operator, namely the $\beta$-Ces\`{a}ro operator \begin{equation} C_\beta(f)(z)=\int_{0}^{z}\frac{f(w)}{w(1-w)^{\beta}}dw \nonumber \end{equation} and its generalization, acting from the $\alpha$-Bloch space to itself, where $f(0)=0$ and $\beta\in\mathbb{R}$. We investigate the boundedness and compactness of the $\beta$-Ces\`{a}ro operators and their generalization. Also we calculate the essential norm and spectrum of these operators.