Inequalities of Hardy-Littlewood-Polya type for functions of operators and their applications

TL;DR

This paper develops generalized inequalities for functions of operators in Hilbert spaces, providing tools for approximation, recovery, and analysis of operators with applications to unbounded operators and element classes.

Contribution

It introduces new multiplicative and additive inequalities for operator functions, extending Hardy-Littlewood-Polya inequalities, and applies them to approximation and recovery problems.

Findings

Derived generalized Hardy-Littlewood-Polya type inequalities

Established bounds for modulus of continuity of operator functions

Solved approximation and recovery problems for unbounded operators

Abstract

In this paper, we derive a generalized multiplicative Hardy-Littlewood-Polya type inequality, as well as several related additive inequalities, for functions of operators in Hilbert spaces. In addition, we find the modulus of continuity of a function of an operator on a class of elements defined with the help of another function of the operator. We then apply the results to solve the following problems: (i) the problem of approximating a function of an unbounded self-adjoint operator by bounded operators, (ii) the problem of best approximation of a certain class of elements from a Hilbert space by another class, and (iii) the problem of optimal recovery of an operator on a class of elements given with an error.