# Stechkin's problem for functions of a self-adjoint operator in a Hilbert space, Taikov-type inequalities and their applications

## TL;DR

This paper addresses the approximation of functionals involving self-adjoint operators in Hilbert spaces, deriving sharp inequalities and exploring applications in operator comparison, differential operators, and spectral measures.

## Contribution

It introduces new sharp Taikov-type inequalities for functions of self-adjoint operators and applies them to operator comparison and spectral analysis.

## Key findings

- Derived sharp Taikov-type inequalities for self-adjoint operators
- Established constants in Hörmander-type operator inequalities
- Extended inequalities to differential operators and spectral measures

## Abstract

In this paper we solve the problem of approximating functionals $(\varphi(A)x, f)$ (where $\varphi(A)$ is some function of self-adjoint operator $A$) on the class of elements of a Hilbert space that is defined with the help of another function $\psi (A)$ of the operator $A$. In addition, we obtain a series of sharp Taikov-type additive inequalities that estimate $|(\varphi(A)x, f)|$ with the help of $\| \psi (A)x\|$ and $\| x\|$. We also present several applications of the obtained results. First, we find sharp constants in inequalities of the type used in H${\rm{\ddot{o}}}$rmander theorem on comparison of operators in the case when operators are acting in a Hilbert space and are functions of a self-adjoint operator. As another application we obtain Taikov-type inequalities for functions of the operator $\frac1i \frac {d}{dt}$ in the spaces $L_2(\RR)$ and $L_2(\TT)$, as well as for integrals with respect to spectral measures, defined with the help of classical orthogonal polynomials.