JournalszaaVol. 25 , No. 4DOI 10.4171/zaa/1301

Composition Operators between <em>H<sup>∞</sup></em> and <em>α</em>-Bloch Spaces on the Polydisc

  • Stevo Stevic

    Serbian Academy of Science, Beograd, Serbia
Composition Operators between <em>H<sup>∞</sup></em> and <em>α</em>-Bloch Spaces on the Polydisc cover

Abstract

Let UnU^n be the unit polydisc of Cn{\mathbb C}^n and \vp(z)=(\vp1(z),,\vpn(z))\vp(z)=(\vp_1(z),\ldots,\vp_n(z)) a holomorphic self-map of Un.U^n. Let H(Un)H(U^n) denote the space of all holomorphic functions on Un,U^n, H(Un)H^\infty(U^n) the space of all bounded holomorphic functions on Un,U^n, and Ba(Un),{\cal B}^a(U^n), a>0,a>0, the aa-Bloch space, i.e.,\hspace{-0.3cm}

Ba(Un)={fH(Un)fBa=f(0)+supzUnk=1nfzk(z)(1zk2)a<+}.{\cal B}^a(U^n)=\bigg\{ f\in H(U^n)\, |\, \|f\|_{{\cal B}^a}=|f(0)|+\sup\limits_{z\in U^n}\sum\limits^n_{k=1} \left|\frac{\partial f} {\partial z_k}(z)\right|\left(1- |z_k|^2\right)^a<+\infty\bigg\}. \hspace{-0.3cm}

We give a necessary and sufficient condition for the composition operator C\vpC_{\vp} induced by \vp\vp to be bounded and compact between H(Un)H^\infty(U^n) and aa-Bloch space Ba(Un),{\cal B}^a(U^n), when a1.a\geq 1.