A note on the strict sigularity of the inclusion between Nakano sequence spaces

Abstract

We characterize the strictly singular inclusions ${\ell }_{p_n}\hookrightarrow {\ell }_{q_n}$ between Nakano sequence spaces providing a useful criterion, namely ${\underline{\lim }}_{n\to \infty }|p_n-q_n|>0$ (also recently obtained by Lang and Nekvinda in [12] with a different proof). It is also noted that no inclusion operator between Nakano sequence spaces is compact, neither $L$-weakly compact nor $M$-weakly compact. An easy criterion is given for the weak compactness of the inclusion.