Arithmetical properties at the level of idempotence

TL;DR

This paper explores extending arithmetic properties like multiplicativity and convolution to operator theory, providing examples relevant to number theory and representing the Euler differential operator through number-theoretic functions and algebraic idempotents.

Contribution

It introduces a novel extension of arithmetic properties to operator theory and offers new representations involving the Euler totient function and algebraic idempotents.

Findings

Extended multiplicativity and convolution concepts to operators

Provided examples connecting number theory and operator theory

Represented the Euler differential operator using number-theoretic functions

Abstract

In this paper we give an attempt to extend some arithmetic properties such as multiplicativity, convolution products to the setting of operators theory. We provide a significant examples which are of interest in number theory. We also give a representation of the Euler differential operator by means of the Euler totient arithmetic function and idempotent elements of some associative unital algebra.