TY - JOUR

T1 - Korovkin-type theorems on B(H) and their applications to function spaces

AU - Bauer, Wolfram

AU - Kumar, V. B. Kiran

AU - Rajan, Rahul

N1 - Funding Information: Open Access funding enabled and organized by Projekt DEAL. Wolfram Bauer wishes to thank Kerala State Higher Education Council for supporting the research visit to Cochin under the Erudite scheme. Rahul Rajan is thankful to IP@Leibniz of Leibniz University of Hannover, Germany for supporting the research visit to Hannover. Rahul Rajan is also supported by UGC-SRF.

PY - 2022/2

Y1 - 2022/2

N2 - 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(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 A2(D) , Fock space F2(C) etc. These results are closely related to the preconditioning problem for operator equations with Toeplitz structure on the unit disk D and on the whole complex plane 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.

AB - 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(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 A2(D) , Fock space F2(C) etc. These results are closely related to the preconditioning problem for operator equations with Toeplitz structure on the unit disk D and on the whole complex plane 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.

KW - Bergman space

KW - Convergence in cluster sense

KW - Fock space

KW - Korovkin-type theorems

KW - Preconditioner

KW - Toeplitz operators

UR - http://www.scopus.com/inward/record.url?scp=85105260428&partnerID=8YFLogxK

U2 - 10.1007/s00605-021-01549-1

DO - 10.1007/s00605-021-01549-1

M3 - Article

AN - SCOPUS:85105260428

VL - 197

SP - 257

EP - 284

JO - Monatshefte für Mathematik

JF - Monatshefte für Mathematik

SN - 0026-9255

IS - 2

ER -