Linear combinations of composition operators from \(H_\alpha^\infty\) to \(H_\beta^\infty\). (Chinese. English summary) Zbl 1538.47047
Summary: For \(i = 1, 2, \dots, M\), let \(\lambda_i\) be nonzero numbers, \(\mathbb{B}\) be the unit ball of the complex space \(\mathbb{C}^N\), \(\varphi_i\) be self-maps of \(\mathbb{B}\), and \(H_\alpha^\infty\) be the weighted holomorphic function space on \(\mathbb{B}\). The compactness of linear combinations of composition operators \(\sum\limits_{i=1}^M \lambda_i C_{\varphi_i}\) from \(H_\alpha^\infty\) to \(H_\beta^\infty\) is discussed in this paper. Moreover, using an equivalent condition of compactness, the author shows that the double difference cancellation \((C_{\varphi_1} - C_{\varphi_2}) - (C_{\varphi_3} - C_{\varphi_1})\) is compact if and only if both \(C_{\varphi_1} - C_{\varphi_2}\) and \(C_{\varphi_3} - C_{\varphi_1}\) are compact.
MSC:
| 47B33 | Linear composition operators |
| 32A37 | Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) |
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