Complex symmetric Toeplitz operators on the generalized derivative Hardy space

Abstract

Abstract The generalized derivative Hardy space S α , β 2 ( D ) $S^{2}_{\alpha ,\beta}(\mathbb{D})$ consists of all functions whose derivatives are in the Hardy and Bergman spaces as follows: for positive integers α, β, S α , β 2 ( D ) = { f ∈ H ( D ) : ∥ f ∥ S α , β 2 2 = ∥ f ∥ H 2 2 + α + β α β ∥ f ′ ∥ A 2 2 + 1 α β ∥ f ′ ∥ H 2 2 < ∞ } , $$ S^{2}_{\alpha ,\beta}(\mathbb{D})= \biggl\{ f\in H(\mathbb{D}) : \Vert {f} \Vert ^{2}_{S^{2}_{ \alpha ,\beta}}= \Vert {f} \Vert ^{2}_{H^{2}}+{ \frac{{\alpha +\beta}}{\alpha \beta}} \bigl\Vert {f'} \bigr\Vert ^{2}_{A^{2}}+ \frac{1}{\alpha \beta} \bigl\Vert {f'} \bigr\Vert ^{2}_{H^{2}}< \infty \biggr\} , $$ where H ( D ) $H({\mathbb{D}})$ denotes the space of all functions analytic on the open unit disk D ${\mathbb{D}}$ . In this paper, we study characterizations for Toeplitz operators to be complex symmetric on the generalized derivative Hardy space S α , β 2 ( D ) $S^{2}_{\alpha ,\beta}(\mathbb{D})$ with respect to some conjugations C ξ $C_{\xi}$ , C μ , λ $C_{\mu , \lambda}$ . 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 ∞ φ ˆ ( − n ) ‾ z ‾ n + ∑ n = 0 ∞ φ ˆ ( n ) z n $\varphi (z)=\sum_{n=1}^{\infty}\overline{\hat{\varphi}(-n)} \overline{z}^{n}+\sum_{n=0}^{\infty}\hat{\varphi}(n)z^{n}$ . Next, we also study complex symmetric Toeplitz operators with non-harmonic symbols on the generalized derivative Hardy space S α , β 2 ( D ) $S^{2}_{\alpha ,\beta}(\mathbb{D})$ .