Property (k- $$\beta )$$ β ) of Musielak–Orlicz and Musielak–Orlicz–Cesàro spaces

Abstract

In this paper, we investigate the geometric property (k- $$\beta )$$ for any fixed integer $$k\ge 1$$ of the space $$l_\varPhi ((E_{n}))$$ generated by a Musielak–Orlicz function $$\varPhi $$ and a sequence $$(E_n)$$ of finite dimensional spaces $$E_{n}$$ , $$n\in {\mathbb {N}}$$ , equipped with both the Luxemburg and the Amemiya norms. As a consequence, we obtain the property (k- $$\beta )$$ of Musielak–Orlicz–Cesaro spaces $$ces_\varPhi $$ using the approach recently considered by Saejung. Some applications to the Cesaro sequence spaces of order $$\alpha $$ and Cesaro difference sequence spaces of order m are also noted.