A Korovkin-type theory for non-self-adjoint Toeplitz operators

We consider the Frobenius optimal preconditioners for Toeplitz operators with complex-valued symbols. We prove Korovkin-type theorems for C⁎-algebra generated by complex-valued continuous periodic symbols corresponding to Toeplitz operators under various modes of convergence in eigenvalue cluster sense. This generalizes the existing results for C⁎-algebra generated by real-valued symbols. In some special cases, the optimal preconditioners can be chosen from matrix algebras with faster convergence rate for the corresponding preconditioned linear systems. We will also consider the linear positive operators associated with the eigenvalues of these preconditioners and obtain a Korovkin-type theorem for such operators as an application of our results.