A canonical form of complex skew-symmetric compact operators and applications to Toeplitz operators

In this paper, we aim to reveal the inherent structure of two important families of operators: the complex skew-symmetric compact operators and the self-commutators of a Toeplitz operator. More specifically, we first completely characterize when a general compact operator is complex skew-symmetric, and use a constructive way to obtain a skew-symmetric canonical form of such an operator. Then we obtain a neat rank-one decomposition of the self-commutators of the Toeplitz operator whose symbol is a bilateral rational function on the Hardy space of the unit disk. As an application of our main results, the complex skew-symmetry of the self-commutators of such a Toeplitz operator is studied. In particular, we completely characterize the corresponding problem for the Toeplitz operator whose symbol is a Laurent polynomial.