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Compact composition operators on the Bloch space. (English) Zbl 0826.47023
Summary: Necessary and sufficient conditions are given for a composition operator $C_\phi f= f\circ \phi$ to be compact on the Bloch space $\cal B$ and on the little Bloch space ${\cal B}_0$. Weakly compact composition operators on ${\cal B}_0$ are shown to be compact. If $\phi\in {\cal B}_0$ is a conformal mapping of the unit disk $\bbfD$ into itself whose image $\phi(\bbfD)$ approaches the unit circle $\bbfT$ only in a finite number of nontangential cusps, then $C_\phi$ is compact on ${\cal B}_0$. On the other hand if there is a point of $\bbfT\cap \overline{\phi(\bbfD)}$ at which $\phi(\bbfD)$ does not have a cusp, then $C_\phi$ is not compact.

47B38Operators on function spaces (general)
47B07Operators defined by compactness properties
30D55H (sup p)-classes (MSC2000)
46J15Banach algebras of differentiable or analytic functions, $H^p$-spaces
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