Composition operators for which C*_φC_φ and C_φ + C*_φ commute

We defineΘ:= {T ∈ L(H): [T*T, T +T*] = 0}. In this paper, we characterize composition operators C_φ and their adjoints C*_φ which belong to Θ, where the maps (Formula presented) are linear fractional selfmaps of the open unit disk D into itself. If φ is an automorphism of D or c = 0, then the case for C_φ ∈ Θis precisely when it is normal. When c = 0, we also prove that if C*_φ ∈ Θ, then either b = 0 or (Formula presented), which implies that the only binormal composition operators C_φ with c = 0 and C*_φ ∈ Θ are normal. Moreover, we show that if φ (0) = 0 and C_φ is not normal, then C_φ ∈ Θ implies that (Formula presented) and d/a is neither real nor purely imaginary, while C*_φ ∈ Θ ensures that (Formula presented) and d/a is real. Finally, we study composition operators C_φ in Θ where φ is an analytic selfmap into D. In particular, this operator has the single-valued extension property.