Complex symmetry of invertible composition operators

A bounded operator T   on a separable Hilbert space HH is said to be complex symmetric   if there exists an orthonormal basis for HH with respect to which T   has a self-transpose matrix representation. In this paper, we study the complex symmetry of composition operators Cϕf=f∘ϕCϕf=f∘ϕ induced on the Hardy space H2H2 by holomorphic self-maps ϕ   of the open unit disk DD. For any holomorphic self-map ϕ   of DD, we establish that if CϕCϕ is complex symmetric, then ϕ   must fix a point in DD. Thus among the automorphisms of DD, only the elliptic ones may induce complex symmetric composition operators. For an elliptic automorphism ϕ, we prove that if ϕ   is not a rotation or of order 3, then CϕCϕ is complex symmetric if and only ifϕ(z)=α−z1−α¯z for some α∈D∖{0}α∈D∖{0}.