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 \(_{*}[A, B]=AB-BA^{*}=0.\) ${}_{\ast }[A,B]=AB-B{A}^{\ast }=0.$ 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.