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Weighted composition operators on the Bargmann–Fock space. (English) Zbl 1171.47021

Let \(dV\) denote the Lebesgue measure on \(\mathbb{C}^N\), \(N \geq 1\). For each \(0<p<\infty\) and \(\alpha>0\), the Bargmann–Fock space \({\mathcal F}_{\alpha}^p(\mathbb{C}^N)\) is defined as
\[ {\mathcal F}_{\alpha}^p(\mathbb{C}^N):= \left \{f \in H(\mathbb{C}^N) : \|f\|_p^p = \left ( \frac{p \alpha}{2 \pi} \right )^N \int_{\mathbb{C}^N} |f(z)|^p e^{-(p \alpha / 2) |z|^2} \; dV(z) < \infty \right \}. \]
In the case \(1 \leq p < \infty\), the space \({\mathcal F}_{\alpha}^p(\mathbb{C}^N)\) is a Banach space with norm \(\|\cdot\|_p\).
Let \(\varphi: \mathbb{C}^N \to \mathbb{C}^N\) be an analytic function and \(\psi \in H(\mathbb{C}^N)\). These maps induce a so-called weighted composition operator
\[ C_{\varphi,\psi}: H(\mathbb{C}^N) \to H(\mathbb{C}^N), \; f \to \psi(f \circ \varphi). \]
In this paper, the author shows that the operator \(C_{\varphi,\psi}: {\mathcal F}_{\alpha}^p(\mathbb{C}^N) \to {\mathcal F}_{\alpha}^p(\mathbb{C}^N)\) is bounded if and only if \(B_{\varphi}^{p, \alpha}(\psi) \in L^{\infty}(\mathbb{C}^N)\), where \[ [B_{\varphi}^{p, \alpha}(\psi))](w) = \left ( \frac{p \alpha}{2 \pi}\right )^N \int_{\mathbb{C}^N} |\psi(z)|^p e^{p \alpha \operatorname{Re} \langle \varphi(z), w\rangle} e^{-(p \alpha /2) (|w|^2 + |z|^2)} \, d V(z) \] for each \(w \in \mathbb{C}^N\). Moreover, in the case \(1 < p < \infty\), it is proved that the essential norm of an operator \(C_{\varphi,\psi}: {\mathcal F}_{\alpha}^p(\mathbb{C}^N) \to {\mathcal F}_{\alpha}^p(\mathbb{C}^N)\) may be estimated in the following way: \[ \limsup_{|w| \to \infty} [B_{\varphi}^{p, \alpha}(\psi)](w) \leq \|C_{\varphi, \psi}\|_e^p \leq C \limsup_{|w| \to \infty} [B_{\varphi}^{p, \alpha}(\psi)](w) \; \]
for some constant \(C>0\).

MSC:

47B33 Linear composition operators
30D15 Special classes of entire functions of one complex variable and growth estimates
47B38 Linear operators on function spaces (general)
46E15 Banach spaces of continuous, differentiable or analytic functions
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