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

TL;DR

This paper characterizes when the inclusion between Nakano sequence spaces is strictly singular, provides a criterion for weak compactness of these inclusions, and notes the non-existence of compact inclusion operators.

Contribution

It offers a new criterion for strict singularity of inclusions between Nakano spaces and discusses compactness properties of these operators.

Findings

Strictly singular inclusion characterized by _{n} - q_{n} > 0

No inclusion operator between Nakano spaces is compact or weakly compact

Criterion for weak compactness of the inclusion provided

Abstract

We characterize the strictly singular inclusions $\ell_{p_n}\hookrightarrow\ell_{q_n}$ between Nakano sequence spaces providing a useful criterion, namely $\varliminf_{n\rightarrow\infty}\vert p_n-q_n\vert>0$ (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.