Semigroups of Hadamard multipliers on the space of real analytic functions

TL;DR

This paper investigates the conditions under which certain multipliers on real analytic functions generate strongly continuous semigroups, focusing on the Euler differential operator of finite order and its role as a generator.

Contribution

It provides new criteria for when multipliers, especially the Euler differential operator, generate strongly continuous semigroups on real analytic functions.

Findings

Euler differential operator of finite order can generate a strongly continuous semigroup under specific conditions.

The paper characterizes when multipliers are generators of semigroups on the space of real analytic functions.

Results clarify the relationship between multiplier properties and semigroup generation in this function space.

Abstract

An operator $M$ acting on the space of real analytic functions is called a multiplier if every monomial is its eigenvector. In this paper we state some results concerning the problem of generating strongly continuous semigroups by multipliers. In particular we show when the Euler differential operator of finite order is a generator and when it is not.