Korovkin-type theorems on B({mathcal {H}}) and their applications to function spaces

Abstract

We prove Korovkin-type theorems in the setting of infinite dimensional Hilbert space operators. The classical Korovkin theorem unified several approximation processes. Also, the non-commutative versions of the theorem were obtained in various settings such as Banach algebras, C^{*}-algebras and lattices etc. The Korovkin-type theorem in the context of preconditioning large linear systems with Toeplitz structure can be found in the recent literature. In this article, we obtain a Korovkin-type theorem on B({mathcal {H}}) which generalizes all such results in the recent literature. As an application of this result, we obtain Korovkin-type approximation for Toeplitz operators acting on various function spaces including Bergman space A^{2}({mathbb {D}}), Fock space F^{2}({mathbb {C}}) etc. These results are closely related to the preconditioning problem for operator equations with Toeplitz structure on the unit disk {mathbb {D}} and on the whole complex plane {mathbb {C}}. It is worthwhile to notice that so far such results are available for Toeplitz operators on circle only. This also establishes the role of Korovkin-type approximation techniques on function spaces with certain oscillation property. To address the function theoretic questions using these operator theory tools will be an interesting area of further research.