The best approximation of closed operators by bounded operators in Hilbert spaces

  • V.F. Babenko Oles Honchar Dnipro National University, 72 Gagarin avenue, 49010, Dnipro, Ukraine
  • N.V. Parfinovych Oles Honchar Dnipro National University, 72 Gagarin avenue, 49010, Dnipro, Ukraine
  • D.S. Skorokhodov Oles Honchar Dnipro National University, 72 Gagarin avenue, 49010, Dnipro, Ukraine
Keywords: best approximation of operators, Stechkin problem, Kolmogorov-type inequalities, self-adjoint operator, Laplace-Beltrami operator, closed operator
Published online: 2022-12-30

Abstract


We solve the problem of the best approximation of closed operators by linear bounded operators in Hilbert spaces under assumption that the operator transforms orthogonal basis in Hilbert space into an orthogonal system. As a consequence, sharp additive Hardy-Littlewood-Pólya type inequality for multiple closed operators is established. We also demonstrate application of these results in concrete situations: for the best approximation of powers of the Laplace-Beltrami operator on classes of functions defined on closed Riemannian manifolds, for the best approximation of differentiation operators on classes of functions defined on the period and on the real line with the weight $e^{-x^2}$, and for the best approximation of functions of self-adjoint operators in Hilbert spaces.

Article metrics
PDF downloads: 435
Abstract views: 697
How to Cite
(1)
Babenko V., Parfinovych N., Skorokhodov D. The Best Approximation of Closed Operators by Bounded Operators in Hilbert Spaces. Carpathian Math. Publ. 2022, 14 (2), 453-463.