Composition Operators between $H^\infty$ and $\alpha$-Bloch Spaces on the Polydisc

Stevo Stevic

doi:10.4171/zaa/1301

JournalszaaVol. 25, No. 4pp. 457–466

Composition Operators between $H^{\infty}$ and $α$ -Bloch Spaces on the Polydisc

Stevo Stevic
Serbian Academy of Science, Beograd, Serbia

$Composition Operators between $H^\infty$ and $\alpha$-Bloch Spaces on the Polydisc cover$

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Abstract

Let $U^{n}$ be the unit polydisc of $C^{n}$ and $φ (z) = (φ_{1} (z), \dots, φ_{n} (z))$ a holomorphic self-map of $U^{n} .$ Let $H (U^{n})$ denote the space of all holomorphic functions on $U^{n},$ $H^{\infty} (U^{n})$ the space of all bounded holomorphic functions on $U^{n},$ and $B^{a} (U^{n}),$ $a > 0,$ the $a$ -Bloch space, i.e.,

B^{a} (U^{n}) = {f \in H (U^{n}) ∣ ∥ f ∥_{B^{a}} = ∣ f (0) ∣ + z \in U^{n} sup k = 1 \sum n \frac{\partial f}{\partial z _{k}} (z) (1 - ∣ z_{k} ∣^{2})^{a} < + \infty} .

We give a necessary and sufficient condition for the composition operator $C_{v a r p hi}$ induced by $φ$ to be bounded and compact between $H^{\infty} (U^{n})$ and $a$ -Bloch space $B^{a} (U^{n}),$ when $a \geq 1.$

Cite this article

Stevo Stevic, Composition Operators between $H^{\infty}$ and $α$ -Bloch Spaces on the Polydisc. Z. Anal. Anwend. 25 (2006), no. 4, pp. 457–466

DOI 10.4171/ZAA/1301