  In a recent article, Dumitru Popa proved an operator version of the Korovkin
theorem. We recall the quantitative version of the Korovkin theorem obtained by
O. Shisha and B. Mond in 1968. In this paper, we obtain a quantitative estimate
for the operator version of the Korovkin theorem obtained by Dumitru Popa. We
also consider various examples where the operator version is applicable and
obtain similar estimates leading to the degree of convergence. In addition, we
obtain the trigonometric analogue of this result by proving the quantitative
version. Finally, we apply this result to the preconditioning problem of large
linear systems with the Toeplitz structure.
