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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.16228 |
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| _version_ | 1866910460487401472 |
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| author | Tong, Cezhong He, Xin Yang, Zicong |
| author_facet | Tong, Cezhong He, Xin Yang, Zicong |
| contents | 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 $\mathcal{B}^α$, 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 $α>1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_16228 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized Volterra-type integral operators between Bloch-type spaces Tong, Cezhong He, Xin Yang, Zicong Functional Analysis 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 $\mathcal{B}^α$, 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 $α>1$. |
| title | Generalized Volterra-type integral operators between Bloch-type spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2405.16228 |