On Hardy-Littlewood-Pólya and Taikov type inequalities for multiple operators in Hilbert spaces

We present a unified approach to obtain sharp mean-squared and multiplicative inequalities of Hardy-Littlewood-Polya and Taikov types for multiple closed operators acting on Hilbert space. We apply our results to establish new sharp inequalities for the norms of powers of the Laplace-Beltrami operators on compact Riemannian manifolds and derive the well-known Taikov and Hardy-Littlewood-Polya inequalities for functions defined on the d-dimensional space in the limit case. Other applications include the best approximation of unbounded operators by linear bounded ones and the best approximation of one class by elements of another class. In addition, we establish sharp Solyar type inequalities for unbounded closed operators with closed range.