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.

我们证明了在无穷维希尔伯特空间算子的环境中的Korovkin型定理。经典的科罗夫金定理统一了几种近似过程。同样，该定理的非可交换形式是在各种设置下获得的，例如Banach代数，\（C ^ {*} \）-代数和格等。在用Toeplitz预处理大型线性系统的情况下，Korovkin型定理结构可以在最近的文献中找到。在本文中，我们在\（B（{\ mathcal {H}}）\）上获得了Korovkin型定理，该定理概括了最近文献中的所有此类结果。作为此结果的应用，我们获得了作用于包括伯格曼空间\（A ^ {2}（{\ mathbb {D}}）\）在内的各种函数空间上的Toeplitz算符的Korovkin型逼近，Fock空间\（F ^ {2}（{\ mathbb {C}}）\）等。这些结果与单元盘\（{\ mathbb {D} } \）以及整个复平面\（{\ mathbb {C}} \）。值得注意的是，到目前为止，这样的结果仅适用于Toeplitz算子。这也确立了Korovkin型逼近技术在具有一定振荡性质的函数空间上的作用。使用这些算子理论工具解决功能理论问题将是进一步研究的一个有趣领域。