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

We characterize the strictly singular inclusions ${\mathrm{\ell }}_{p_n}\hookrightarrow {\mathrm{\ell }}_{q_n}$ between Nakano sequence spaces providing a useful criterion, namely ${\mathrm{}}_{n\to \mathrm{\infty }}\mathrm{\mid }{p}_n-q_n\mathrm{\mid }>0$ (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.