## 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 $$\mathcal B$$ and on the little Bloch space $${\mathcal B}_0$$. Weakly compact composition operators on $${\mathcal B}_0$$ are shown to be compact. If $$\phi\in {\mathcal B}_0$$ is a conformal mapping of the unit disk $$\mathbb{D}$$ into itself whose image $$\phi(\mathbb{D})$$ approaches the unit circle $$\mathbb{T}$$ only in a finite number of nontangential cusps, then $$C_\phi$$ is compact on $${\mathcal B}_0$$. On the other hand if there is a point of $$\mathbb{T}\cap \overline{\phi(\mathbb{D})}$$ at which $$\phi(\mathbb{D})$$ does not have a cusp, then $$C_\phi$$ is not compact.

### MSC:

 47B38 Linear operators on function spaces (general) 47B07 Linear operators defined by compactness properties 30D55 $$H^p$$-classes (MSC2000) 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces

### Keywords:

composition operator; Bloch space; nontangential cusps
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### References:

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