Normal structure and weakly normal structure of Orlicz sequence spaces

Abstract:

For a convex Orlicz function $\varphi :{{\bf {R}}_ + } \to {{\bf {R}}_ + } \cup \{ \infty \}$ and the associated Orlicz sequence space ${l_\varphi }$, we consider the following five properties: (1) ${l_\varphi }$ has a subspace isometric to ${l_1}$. (2) ${l_\varphi }$ is Schur. (3) ${l_\varphi }$ has normal structure. (4) Every weakly compact subset of ${l_\varphi }$ has normal structure. (5) Every bounded sequence in ${l_\varphi }$ has a subsequence $({x_n})$ which is pointwise and almost convergent to $x \in {l_\varphi }$, i.e., $\lim {\sup _{n \to \infty }}\parallel {x_n} - x{\parallel _{\varphi }} < \lim \inf _{n \to \infty }\parallel {x_n} - y{\parallel _\varphi }$ for all $y \ne x$. Our results are: (1) $\Leftrightarrow \;\varphi$ is either linear at $0\;(\varphi (s)/s = c > 0,0 < s \leqslant t)$ or does not satisfy the ${\Delta _2}$-condition at $0$. (2) $\Leftrightarrow \;{l_\varphi }$ is isomorphic to ${l_1}\; \Leftrightarrow \;\varphi ’(0) = {\lim _{t \to 0}} \varphi (t)/t > 0$. (3) $\Leftrightarrow \varphi$ satisfies the ${\Delta _2}$-condition at $0, \varphi$ is not linear at $0$ and $C(\varphi ) = \sup \{ \varphi (t) < 1\} > \frac {1}{2}$. (4) $\Leftrightarrow \varphi$ satisfies the ${\Delta _2}$-condition at $0$ and $C (\varphi ) > \frac {1}{2}\;{\rm {or}}\;\varphi ’(0) > 0$. (5) $\Leftrightarrow \;\varphi$ satisfies the ${\Delta _2}$-condition at $0$ and $C(\varphi ) = 1$. The last equivalence contains a result of Lami-Dozo [10].