Norms of some operators from Bergman spaces to weighted and Bloch-type spaces. (English) Zbl 1160.47027

Stević, Stevo

Util. Math. 76, 59-64 (2008).

Let \(B\) be the open unit ball in \(\mathbb C^n\), \(n \in \mathbb N\). The weighted Bergman spaces \(A_{\alpha}^p\) are defined by
\[ A_{\alpha}^p:= \{ f \in H(B); \; \| f\| _{A_{\alpha}^p}:= \int_B | f(z)| c_{\alpha} (1-| z| ^2)^{\alpha} \; dV(z) < \infty \},\quad p>1,\quad \alpha > -1, \]
where \(dV(z)\) is the Lebesgue measure and \(c_{\alpha}\) chosen so that \( \int_B c_{\alpha} (1-| z| ^2)^{\alpha} \; dV(z) =1\). For \(u \in H(B)\) and an analytic self-map \(\varphi\) of \(B\), the corresponding weighted composition operator is defined by
\[ u C_{\varphi}: H(B) \to H(B), \; f \to u (f \circ \varphi). \]
In this paper, the authors determines the norm of such a weighted composition operator acting between the weighted Bergman space \(A_{\alpha}^p\) and the weighted Banach space of holomorphic functions
\[ H_{\mu}^{\infty}:= \{ f \in H(B) \mid \| f\| _{\mu}:= \sup_{z \in B} \mu(z) | f(z)| < \infty \}. \]
In the second part of the article, the author restricts himself to the setting \(n=1\). Let \(\varphi\) be an analytic self-map of the open unit disk \(\mathbb D\), and \(g \in H(\mathbb D)\). For \(f \in H(\mathbb D)\), the products of Volterra and composition operators are given by \[ C_{\varphi} J_g f(z):= \int_{0}^{\varphi(z)} f(\xi) g(\xi) \; d \xi \text{ and } J_g C_{\varphi} f(z):= \int_0^z f(\varphi(\xi)) g(\xi) \; d \xi. \] The norms of such operators acting between weighted Bergman spaces \(A_{\alpha}^p\) and weighted Bloch type spaces \(B_{\mu}\) are computed.

Reviewer: Elke Wolf (Paderborn)

MSC:

47B33	Linear composition operators
47B38	Linear operators on function spaces (general)
46E15	Banach spaces of continuous, differentiable or analytic functions

Keywords:

weighted composition operators; Volterra operators; Bergman spaces; Bloch spaces; weighted type spaces