Bilichenko R. Inequalities of Kolmogorov type for operators.

The object is the inequalities of Hardy-Littlwood-Polia, Taikov, Shadrin type for self-adjoint and normal operators acting in Hilbert space. The aim is the esteblishment of exact inequalities for powers of arbitrary self-adjoint and normal operators acting in Hilbert space; the solving of the approximation theory problems which related in these exact ineaualities. Methods are the general methods of solving of the approximation theory extremal problems, the methods of proof the inequalities of Kolmogorov type, the estimate methods best approximation of unbounded operators by bounded, and general facts or the functional analysis and the theory of functions. New exact inequalities for powers of self-adjoint and normal operators acting in Hilbert space, in particular inequalities that estimate the value of unbounded arbitrary functional on the application of an element to a power operator are establishing, including in the case of compositions of degrees pairwise jumping self-adjoint operators. For self-adjoint and normal operators acting in a Hilbert space the solutions of Styechkin's problem the approximation of unbounded operators by bounded; the problem of finding the continuity modulus for arbitrary power of operator; the problem of approximation of unbounded functional by bounded; the problem of approximation the class that is given powers operator by homotet of other class; the problem of restoring the values of operator on class elements defined with an error are solved. Scope - the approximation theory, functional analysis, the learning process.