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

Abstract

Let $\mathbf{\Phi }=({\varphi }_n)$ 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 $l_{\mathbf{\Phi }}^A(X)$ 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 $l_{\mathbf{\Phi }}^A(X)$ 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 $l_{\mathbf{\Phi }}^A$ and operator ideals under certain conditions on the matrix $A$ are discussed.

18 pages