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

TL;DR

This paper extends Korovkin-type theorems to infinite-dimensional operators, analyzing convergence of completely positive maps and their applications to preconditioning large Toeplitz-structured linear systems.

Contribution

It develops new Korovkin-type theorems for operator convergence in infinite dimensions and applies them to analyze preconditioners for large Toeplitz matrices.

Findings

Convergence modes for matrix sequences are extended to infinite-dimensional operators.

Asymptotic behavior of preconditioners is characterized using completely positive maps.

Limit points of preconditioner sequences are equivalent modulo compact operators.

Abstract

The classical as well as non commutative Korovkin-type theorems deal with convergence of positive linear maps with respect to modes of convergences such as norm convergence and weak operator convergence. In this article, Korovkin-type theorems are proved for convergence of completely positive maps with respect to weak, strong and uniform clustering of sequences of matrices of growing order. Such modes of convergence were originally considered for Toeplitz matrices (see [23],[26]). As an application, we translate the Korovkin-type approach used in the finite dimensional case, in the setting of preconditioning large linear systems with Toeplitz structure, into the infinite dimensional context of operators acting on separable Hilbert spaces. The asymptotic of these pre-conditioners are obtained and analyzed using the concept of completely positive maps. It is observed that any two limit points of 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.