The classical as well as noncommutative Korovkin-type theorems deal with the convergence of positive linear maps with respect to different modes of convergence, like norm or weak operator convergence etc. In this article, new versions of Korovkin-type theorems are proved using the notions of convergence induced by strong, weak and uniform eigenvalue clustering of matrix sequences with growing order. Such modes of convergence were originally considered for the special case of Toeplitz matrices and indeed the Korovkin-type approach, in the setting of preconditioning large linear systems with Toeplitz structure, is well known. Here we extend this finite-dimensional ap- proach to the infinite-dimensional context of operators acting on separable Hilbert spaces. The asymptotics of these preconditioners are evaluated and analyzed using the concept of completely positive maps. It is observed that any two limit points, under Kadison' BW-topology, of the same sequence of preconditioners are equal modulo compact opera- tors. Moreover, this indicates the role of preconditioners in the spectral approximation of bounded self-adjoint operators.

Preconditioners and Korovkin-type theorems for infinite-dimensional bounded linear operators via completely positive maps

SERRA CAPIZZANO, STEFANO
2013-01-01

Abstract

The classical as well as noncommutative Korovkin-type theorems deal with the convergence of positive linear maps with respect to different modes of convergence, like norm or weak operator convergence etc. In this article, new versions of Korovkin-type theorems are proved using the notions of convergence induced by strong, weak and uniform eigenvalue clustering of matrix sequences with growing order. Such modes of convergence were originally considered for the special case of Toeplitz matrices and indeed the Korovkin-type approach, in the setting of preconditioning large linear systems with Toeplitz structure, is well known. Here we extend this finite-dimensional ap- proach to the infinite-dimensional context of operators acting on separable Hilbert spaces. The asymptotics of these preconditioners are evaluated and analyzed using the concept of completely positive maps. It is observed that any two limit points, under Kadison' BW-topology, of the same sequence of preconditioners are equal modulo compact opera- tors. Moreover, this indicates the role of preconditioners in the spectral approximation of bounded self-adjoint operators.
2013
Completely positive maps; Frobenius norm; Preconditioners
Kumar, K.; Namboodiri, M. N. N.; SERRA CAPIZZANO, Stefano
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/1927321
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