The skew commutators of Toeplitz operators or Hankel operators on Hardy spaces

Let A and B be two bounded linear operators on a Hilbert space. B is called the skew commutator of A if ∗[A,B]=AB-BA∗=0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{*}[A, B]=AB-BA^{*}=0.$$\end{document} In this paper, we completely characterize when a Toeplitz operator on the Hardy space is a skew commutator of a Hankel operator and when a Hankel operator on the Hardy space is a skew commutator of a Toeplitz operator. Moreover, we also obtain a necessary and sufficient condition for the product of a Hankel operator and a Toeplitz operator to be self-adjoint on the Hardy space.