Boundedness From Below of Multiplication Operators Between $alpha$-Bloch Spaces

Canadian mathematical bulletin

关键词: $alpha$-Bloch function;    multiplication operator;   

DOI  :  10.4153/CMB-2010-007-5

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

In this paper, the boundedness from below of multiplicationoperators between $alpha$-Bloch spaces $mathcal B^alpha$, $alphagt 0$, on theunit disk $D$ is studied completely. For a bounded multiplicationoperator $M_ucolon mathcal B^alphaomathcal B^eta$, defined by $M_uf=uf$ for$finmathcal B^alpha$, we prove the following result:(i) If $0lt etalt alpha$, or$0lt alphale1$ and $alphalt eta$, $M_u$ is not bounded below;(ii) if $0lt alpha=etale1$, $M_u$ is bounded below if and only if$liminf_{zopartial D}|u(z)|gt 0$;(iii) if $1lt alphaleeta$, $M_u$ isbounded below if and only if there exist a $deltagt 0$ and a positive$rlt 1$ such that for every point $zin D$ there is a point $z'inD$ with the property $d(z',z)lt r$ and$(1-|z'|^2)^{eta-alpha}|u(z')|gedelta$, where $d(cdot,cdot)$ denotesthe pseudo-distance on $D$.

【 授权许可】

Unknown