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

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