Generalized Volterra-type integral operators between Bloch-type spaces

TL;DR

This paper introduces a generalized class of Volterra-type integral operators acting on Bloch-type spaces, analyzing their boundedness, compactness, and rigidity, and establishing conditions under which these properties hold.

Contribution

The paper extends existing integral operators to a more general form on Bloch-type spaces and characterizes their boundedness, compactness, and rigidity properties.

Findings

Boundedness and compactness of the sum of operators are equivalent to those of individual operators.

The boundedness and compactness of certain operators are independent of the iteration number when >1.

The introduced operators unify and generalize previous integral operators in complex analysis.

Abstract

The Volterra-type integral operator plays an essential role in modern complex analysis and operator theory. Recently, Chalmoukis \cite{Cn} introduced a generalized integral operator, say $I_{g,a}$, defined by $$I_{g,a}f=I^n(a_0f^{(n-1)}g'+a_1f^{(n-2)}g''+\cdots+a_{n-1}fg^{(n)}),$$ where $g\in H(\mathbb{D})$ and $a=(a_0,a_1,\cdots,a_{n-1})\in \mathbb{C}^n$. $I^n$ is the $n$th iteration of the integral operator $I$. In this paper, we introduce a more generalized integral operators $I_{\mathbf{g}}^{(n)}$ that cover $I_{g,a}$ on the Bloch-type space $\mathcal{B}^{\alpha}$, defined by $$I_{\mathbf{g}}^{(n)}f=I^n(fg_0+\cdots+f^{(n-1)}g_{n-1}).$$ We show the rigidity of the operator $I_{\mathbf{g}}^{(n)}$ and further the sum $\sum_{i=1}^nI_{g_i}^{N_i,k_i}$, where $I_{g_i}^{N_i,k_i}f=I^{N_i}(f^{(k_i)}g_i)$. Specifically, the boundedness and compactness of $\sum_{i=1}^nI_{g_i}^{N_i,k_i}$ are equal to those of each $I_{g_i}^{N_i,k_i}$. Moreover, the boundedness and compactness of $I^n((fg')^{(n-1)})$ are independent of $n$ when $\alpha>1$.