Complex symmetric Toeplitz operators on the generalized derivative Hardy space

The generalized derivative Hardy space Sα,β2(D) consists of all functions whose derivatives are in the Hardy and Bergman spaces as follows: for positive integers α, β, Sα,β2(D)={f∈H(D):∥f∥Sα,β22=∥f∥H22+α+βαβ∥f′∥A22+1αβ∥f′∥H22<∞}, where H(D) denotes the space of all functions analytic on the open unit disk D. In this paper, we study characterizations for Toeplitz operators to be complex symmetric on the generalized derivative Hardy space Sα,β2(D) with respect to some conjugations C_ξ, C_μ,λ. Moreover, for any conjugation C, we consider the necessary and sufficient conditions for complex symmetric Toeplitz operators with the symbol φ of the form φ(z)=∑n=1∞φˆ(−n)‾z‾n+∑n=0∞φˆ(n)zn. Next, we also study complex symmetric Toeplitz operators with non-harmonic symbols on the generalized derivative Hardy space Sα,β2(D).