Complex symmetric Toeplitz operators on the generalized derivative Hardy space

Abstract

The generalized derivative Hardy space 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, Sα,β2(D)={f∈H(D):∥f∥Sα,β22=∥f∥H22+α+βαβ∥f′∥A22+1αβ∥f′∥H22<∞},\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ S^{2}_{\\alpha ,\\beta}(\\mathbb{D})= \\biggl\\{ f\\in H(\\mathbb{D}) : \\Vert {f} \\Vert ^{2}_{S^{2}_{ \\alpha ,\\beta}}= \\Vert {f} \\Vert ^{2}_{H^{2}}+{ \\frac{{\\alpha +\\beta}}{\\alpha \\beta}} \\bigl\\Vert {f'} \\bigr\\Vert ^{2}_{A^{2}}+ \\frac{1}{\\alpha \\beta} \\bigl\\Vert {f'} \\bigr\\Vert ^{2}_{H^{2}}< \\infty \\biggr\\} , $$\\end{document} where H({mathbb{D}}) denotes the space of all functions analytic on the open unit disk {mathbb{D}}. In this paper, we study characterizations for Toeplitz operators to be complex symmetric on the generalized derivative Hardy space S^{2}_{alpha ,beta}(mathbb{D}) with respect to some conjugations C_{xi}, 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 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^{2}_{alpha ,beta}(mathbb{D}).