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Products of composition, multiplication and radial derivative operators between Banach spaces of holomorphic functions on the unit ball. (English) Zbl 1523.47041

Let \(H(\mathbb{B})\) be the space of holomorphic functions on the open unit ball \(\mathbb{B}\) in the \(n\)-dimensional Euclidean space \(\mathbb{C}^n\), and \(S(\mathbb{B})\) be the class of holomorphic self-maps of \(\mathbb{B}\). Let \(\mathcal{R}\) be the radial derivative operator on \(H(\mathbb{B})\); i.e., \[ \mathcal{R}f(z):=\sum_{j=1}^nz_j\frac{\partial f}{\partial z_j}(z) \] for all \(f\in H(\mathbb{B})\). For any \(\varphi\in S(\mathbb{B})\) and \(\psi\in H(\mathbb{B})\), let \(C_\varphi:f\mapsto f\circ\varphi\) and \(M_\psi:f\mapsto\psi.f\) be the composition and multiplication operators induced by \(\varphi\) and \(\psi\), respectively. Given \(\varphi\in S(\mathbb{B})\) and \(\psi_1,\psi_2,\psi_3\in H(\mathbb{B})\), the authors characterize the boundedness, compactness and order boundedness of the operator \[ \textbf{T}_{\psi_1,\psi_2,\psi_3,\varphi}:=M_{\psi_1}C_\varphi+M_{\psi_2}C_\varphi\mathcal{R}+M_{\psi_3}\mathcal{R}C_\varphi \] mapping from a large class of Banach spaces \(X\) of holomorphic functions on \(\mathbb{B}\) into certain weighted-type spaces. They also give the norm estimate and essential norm estimate of such an operator. Furthermore, they obtain similar characterizations and estimates for the weighted composition operator \(M_{\psi}C_\varphi\) mapping from \(X\) into the Bloch-type space \(\mathcal{B}_\mu\) (or \(\mathcal{B}_{\mu,0}\)). Their results show that the obtained characterizations and estimates of these operators depend only on the symbols and the norm of the point-evaluation functionals on the domain space.

MSC:

47B33 Linear composition operators
47B38 Linear operators on function spaces (general)
46E15 Banach spaces of continuous, differentiable or analytic functions
47B65 Positive linear operators and order-bounded operators
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References:

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