$M$-ideal properties in Orlicz-Lorentz spaces

TL;DR

This paper characterizes the dual space and $M$-ideal properties of Orlicz-Lorentz spaces, providing explicit formulas for bounded linear functionals and analyzing the impact of different norms on these properties.

Contribution

It offers explicit formulas for dual norms in Orlicz-Lorentz spaces and establishes $M$-ideal properties of the order-continuous subspace under Luxemburg norm.

Findings

Norm formulas for bounded linear functionals derived

Order-continuous subspace is an $M$-ideal with Luxemburg norm

$M$-ideal property fails for Orlicz norm when $ exists ext{ appropriate } riangle_2$ condition

Abstract

We provide explicit formulas for the norm of bounded linear functionals on Orlicz-Lorentz function spaces $\Lambda_{\varphi,w}$ equipped with two standard Luxemburg and Orlicz norms. Any bounded linear functional is a sum of regular and singular functionals, and we show that the norm of a singular functional is the same regardless of the norm in the space, while the formulas of the norm of general functionals are different for the Luxemburg and Orlicz norm. The relationship between equivalent definitions of the modular $P_{\varphi,w}$ generating the dual space to Orlicz-Lorentz space is discussed in order to compute the norm of a bounded linear functional on $\Lambda_{\varphi,w}$ equipped with Orlicz norm. As a consequence, we show that the order-continuous subspace of Orlicz-Lorentz space equipped with the Luxemburg norm is an $M$-ideal in $\Lambda_{\varphi,w}$, while this is not true for the space with the Orlicz norm when $\varphi$ is an Orlicz $N$-function not satisfying the appropriate $\Delta_2$ condition. The analogous results on Orlicz-Lorentz sequence spaces are given.