2
$\begingroup$

Let $1<p<\infty$. The James $p$-space $J_{p}$ is the Banach space of all sequences of real numbers $(a_{i})_{i}\in c_{0}$ such that $$\|(a_{i})_{i}\|=\sup\{(\sum_{j=1}^{n}|a_{p_{j-1}}-a_{p_{j}}|^{p})^{\frac{1}{p}}:p_{0}<p_{1}<\cdots<p_{n}, n\in\mathbb{N}\}<\infty.$$ Let $(e_{n})_{n}$ be the unit vector basis of $J_{p}$ and $(e^{*}_{n})_{n}$ be the sequence of biorthogonal functionals. It is known that $(e_{n})_{n}$ is a shrinking monotone basis for $J_{p}$. I have the following two questions:

Question 1: Let $z^{*}_{n}=\sum_{i=k_{n-1}+1}^{k_{n}}c_{i}e^{*}_{i}$ be a semi-normalized block basic sequence in $J^{*}_{p}$ and suppose that $\sum_{i=k_{n-1}+1}^{k_{n}}c_{i}=0$ for each $n\in \mathbb{N}$. Is the sequence $(z^{*}_{n})_{n}$ equivalent to the unit vector basis of $l_{p}$?

Question 1 is true for $p=2$ as proved by Alfred Andrew in Israel J. Math.(1981).

Question 2: Is the dual $J_{p}^{*}$ of $J_{p}$ weakly sequentially complete?

Thank you!

$\endgroup$
3
  • 2
    $\begingroup$ Q 1: Doesn't Andrew's proof show that $z_n^*$ is equivalent to the unit vector basis of $\ell_q$, where $q$ is the conjugate index to $p$? $\endgroup$ May 1, 2016 at 0:01
  • 2
    $\begingroup$ Q 2: It is elementary that a weakly sequentially complete space that has separable dual is reflexive. $\endgroup$ May 1, 2016 at 0:02
  • $\begingroup$ Q 1: I am not sure that Andrew's proof can be used to show that $(z^{*}_{n})_{n}$ is equivalent to the unit vector basis of $l_{q}$. I have to check it. $\endgroup$ May 1, 2016 at 0:14

1 Answer 1

0
$\begingroup$

I think that this question should be considered as answered, following the comments of Bill Johnson as:

(1) One can follow the argument of Andrew and show that $z_n^*$ is equivalent to the unit vector basis of $\ell _q$, where $q$ is conjugate to $p$.

(2) The well-known result of Rosenthal implies that each bounded sequence either contains a weakly Cauchy subsequence or a subsequence equivalent to the unit vector basis of $\ell_ 1$ Hence, by the Eberlein-Smulian theorem, if the space is weakly complete, it should either contain $\ell_1$ or be reflexive. (See Lindenstrauss and Tzafriri, vol. 1, page 99 for Rosenthal's theorem.)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.