On some geometric properties of generalized Musielak-Orlicz sequence space and corresponding operator ideals

TL;DR

This paper introduces a generalized Musielak-Orlicz sequence space linked to an infinite matrix and explores its geometric properties, completeness, and operator ideals, extending the understanding of these spaces in Banach space theory.

Contribution

It defines a new vector-valued Musielak-Orlicz sequence space associated with a matrix and investigates its geometric properties, completeness, and related operator ideals.

Findings

The space is a complete normed linear space under certain conditions.

It is $\sigma$-Dedekind complete when the Banach space is so.

The space exhibits properties like uniform monotonicity and the uniform Opial property.

Abstract

Let $\bold{\Phi}=(\phi_n)$ be a Musielak-Orlicz function, $X$ be a real Banach space and $A$ be any infinite matrix. In this paper, a generalized vector-valued Musielak-Orlicz sequence space $l_{\bold {\Phi}}^{A}(X)$ is introduced. It is shown that the space is complete normed linear space under certain conditions on the matrix $A$. It is also shown that $l_{\bold{\Phi}}^{A}(X)$ is a $\sigma$- Dedikind complete whenever $X$ is so. We have discussed some geometric properties, namely, uniformly monotone, uniform Opial property for this space. Using the sequence of $s$-number (in the sense of Pietsch), the operators of $s$-type $l_{\bold{\Phi}}^{A}$ and operator ideals under certain conditions on the matrix $A$ are discussed.