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The classical as well as non commutative Korovkin-type theorems deal withconvergence of positive linear maps with respect to modes of convergences suchas norm convergence and weak operator convergence. In this article,Korovkin-type theorems are proved for convergence of completely positive mapswith respect to weak, strong and uniform clustering of sequences of matrices ofgrowing order. Such modes of convergence were originally considered forToeplitz matrices (see [23],[26]). As an application, we translate theKorovkin-type approach used in the finite dimensional case, in the setting ofpreconditioning large linear systems with Toeplitz structure, into the infinitedimensional context of operators acting on separable Hilbert spaces. Theasymptotic of these pre-conditioners are obtained and analyzed using theconcept of completely positive maps. It is observed that any two limit pointsof the same sequence of pre-conditioners are the same modulo compact operators.Finally, we prove the generalized versions of the Korovkin type theorems in[23]. Keywords: Completely positive maps, Frobenius norm, Pre-conditioners.

Korovkin results and Frobenius optimal approximants for infinite dimensional bounded linear operators