Boundedness From Below of Multiplication Operators Between α-Bloch Spaces

Chen, Huaihui ; Zhang, Minzhu

Canadian mathematical bulletin, Tome 53 (2010) no. 1, pp. 23-36

Résumé

In this paper, the boundedness from below of multiplication operators between $\alpha$ -Bloch spaces ${{B}^{\alpha }},\,\alpha \,>\,0$ , on the unit disk $D$ is studied completely. For a bounded multiplication operator ${{M}_{u}}\,:\,{{B}^{\alpha }}\,\to \,{{B}^{\beta }}$ , defined by ${{M}_{u}}f\,=\,uf$ for $f\,\in \,{{B}^{\alpha }}$ , we prove the following result:(i) If $0<\beta <\alpha ,\,\text{or}\,\text{0}<\alpha \le \text{1}\,\text{and}\,\alpha <\beta \text{,}\,{{M}_{u}}$ is not bounded below;(ii) if $0\,<\,\alpha \,=\,\beta \,\le \,1,\,{{M}_{u}}$ is bounded below if and only if lim ${{\inf }_{z\to \partial D}}\,\left| u\left( z \right) \right|\,>\,0;$ (iii) if $1\,<\,\alpha \,\le \,\beta ,\,{{M}_{u}}$ is bounded below if and only if there exist a $\delta \,>\,0$ and a positive $r\,<\,1$ such that for every point $z\,\in \,D$ there is a point ${{z}^{'}}\,\in \,D$ with the property $d\left( {{z}^{'}},\,z \right)\,<\,r$ and ${{\left( 1\,-\,{{\left| {{z}^{'}} \right|}^{2}} \right)}^{\beta -\alpha }}\left| u\left( {{z}^{'}} \right) \right|\,\ge \,\delta$ , where $d\left( \cdot ,\,\cdot\right)$ denotes the pseudo-distance on $D$ .

Détail

DOI : 10.4153/CMB-2010-007-5

Mots-clés : 32A18, 30H05, α-Bloch function, multiplication operator