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The composition operator $C_{ϕ_a}f=f\circϕ_a$ on the Hardy-Hilbert space $H^2(\mathbb{D})$ with affine symbol $ϕ_a(z)=az+1-a$ and $0<a<1$ has the property that every bounded linear operator on a separable complex Hilbert space has a proper closed invariant subspace if and only if every minimal invariant subspace for $C_{ϕ_a}$ is one-dimensional.
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