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

Abstract

Let ${\Phi} =(\phi_n)$ 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 $l_{{\Phi}}^{A}(X)$ is introduced. It is shown that the space is a complete normed linear space under certain conditions on the matrix $A$. It is also shown that $l_{{\Phi}}^{A}(X)$ is a $\sigma$-Dedekind 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 $l_{{\Phi}}^{A}$ and operator ideals under certain conditions on the matrix $A$ are discussed.