Let $E$ be a separable Banach space with symmetric basis $\{e\_i\}$ (it is also called a symmetric sequence space).
Let $\{x\_i\}$ be a normalized disjoint sequence in $E$, i.e.,
$\lVert x\_i\rVert\_E=1$ and $x\_i =\sum\_{n=n\_i}^{n\_{i+1}-1}a\_n e\_n $ for some strictly increasing sequence $\{n\_i\}$.


Assume that the uniform norm $\lVert x\_i\rVert\_\infty \to 0$ as $i\to \infty$ ($\lVert x\rVert\_\infty=\sup \lvert a\_n\rvert$, $x =\sum a\_n e\_n $). Assume, in addition, that $\{x\_i\}$ is a symmetric basic sequence in $E$.
Then, I guess, if $\{x\_i\}$ is equivalent to $\{e\_i\}$, then $E$ is equivalent to $\ell\_p$ for some $p\in [1,\infty)$ or it is equivalent to $c\_0$.


Such a property holds true for $\ell\_p$ but I don't know any other symmetric sequence space having such a property.



On the symmetric basic sequence of a symmetric sequence space
The question is not well formulated but i will answer the way I understood it. I think you are asking if there is any space with a symmetric basis other than $c\_0, \ell\_p$ which contains symmetric basic sequences with sup norm tends to zero and equivalent to the basis. The answer is yes. For instance, a minimal Orlicz sequence spaces $\ell\_M$ not isomorphic to $\ell\_p$ have such symmetric sequences, in fact, they are some constant coefficient blocks (with increasing support).


Minimal Orlicz sequence space means something else than a minimal Banach space. If you are not familiar with Orlicz space terminology you will need to skim through Chapter 4 of Lindenstaruss-Tzafriri's book. Here is a very quick pointers: For each $\ell\_M$ there is associated set $E\_{M,1}$ of Orlicz functions. If $N\in E\_{M,1}$ then $\ell\_N\subseteq \ell\_M$ and moreover the basis of $\ell\_N$ is some constant coefficient block basis. $\ell\_M$ minimal means for every $N\in E\_{M,1}$ we have $E\_{N\_,1}=E\_{M,1}$. So in particular, if $E\_{M,1}\neq \{t^p\}$, in such a space every constant coefficient blocks basis have a further blocks sequence which is equivalent to the basis. It is clear that the sup norm of such a sequence tends to zero.


