For affine φ(w)=aw+b on H^2(C+), Cφ is complex symmetric iff a=1 or Re(b)=0, cyclic iff a≥1 and Re(b)>0, and never hypercyclic.
Matache, Invertible and normal composition operatos on the Hilbert H ardy space of a half-plane, Concr
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Complex symmetry and cyclicity of composition operators on $H^2(\mathbb{C}_+)$
For affine φ(w)=aw+b on H^2(C+), Cφ is complex symmetric iff a=1 or Re(b)=0, cyclic iff a≥1 and Re(b)>0, and never hypercyclic.