On the operators of Hardy-Littlewood-P\'olya type

TL;DR

This paper introduces new Hardy-Littlewood-Pólya-type operators, analyzes their boundedness and compactness, and calculates their exact norms in specific cases, expanding the understanding of these operators in functional analysis.

Contribution

It presents novel Hardy-Littlewood-Pólya-type operators induced by measures, with conditions for boundedness and compactness, and computes their norms in special scenarios.

Findings

Established necessary and sufficient conditions for boundedness.

Determined criteria for compactness of the operators.

Calculated exact operator norms in particular cases.

Abstract

In this paper we introduce and study several new Hardy-Littlewood-P\'olya-type operators. In particular, we study a Hardy-Littlewood-P\'olya-type operator induced by a positive Borel measure on $[0,1)$. We establish some sufficient and necessary conditions for the boundedness (compactness) of these operators. We also determine the exact values of the norms of the Hardy-Littlewood-P\'olya-type operators for certain special cases.