For any given n-by-n matrix A_n, a specific circulant preconditioner t_F(A_n) proposed by E. Tyrtyshnikov [SIAM J. Matrix Anal. Appl., Vol. 13 (1992), pp. 459–473] is defined to be the solution of  over all n-by-n nonsingular circulant matrices C_n. The t_F(A_n), called the superoptimal circulant preconditioner, has been proved to be a good preconditioner for a large class of structured systems including some ill-conditioned problems from image processing. In this paper, we study this preconditioner from an operator viewpoint. We will give some relationships between the optimal preconditioner (operator) proposed by T. Chan [SIAM J. Sci. Statist. Comput., Vol. 9 (1988), pp. 766–771] and superoptimal preconditioner (operator).