2
$\begingroup$

Let $\{e_n,e^*_n\}_{n=1}^\infty$ be a biorthogonal system in $\mathbb{X}\times\mathbb{X}^*$. The system $\{e_n\}_{n=1}^\infty$ is called a basic sequence if it is a Schauder basis of the subspace $E={\overline{{\rm span} \,\{e_n\}_{n=1}^\infty}}^{\mathbb X}$. A natural sufficient condition is to test whether the projection operators $$ S_N x=\sum_{n=1}^N e^*_n(x)e_n, \quad x\in\mathbb X,$$ are uniformly bounded in $\mathbb X$.

Is this condition also necessary? That is, is it true that $$ \{e_n\}_{n=1}^\infty \mbox{ is a basic sequence in $\mathbb X$ } \quad\Longleftrightarrow \quad\sup_N\|S_N\|_{\mathbb X\to\mathbb X}<\infty\;? \tag{*}$$ This is the case in simple examples, such as when $\{e_n\}$ is the canonical system in $\ell_\infty$ or the Haar system in $L_\infty$.

If the answer is NO, are there natural additional hypothesis on $\mathbb X$ (or on $\{e_n,e^*_n\}$) that ensure that ($*$) holds?

$\endgroup$
1
$\begingroup$

I think that I found an easy negative answer by adding a 1-dimensional subspace (sorry for not thinking enough before posting).

The details would be as follows. Assume that $\{e_n\}_{n=1}^\infty$ is a Schauder basis in $\mathbb X$, with dual functionals $e^*_n$, and say normalized $\|e_n\|_{\mathbb X}=\|e^*_n\|_{\mathbb X^*}=1$. Define a new space $\widehat{\mathbb {X}}=\mathbb {X}\oplus_1[x_0]$, with norm

$$\|(x,\lambda x_0)\|_{\widehat{\mathbb{X}}}=\|x\|_{\mathbb X}+|\lambda|.$$

Define a new biorthogonal system $\{\hat{e}_n,\hat{e}^*_n\}_{n=1}^\infty$ in $\widehat{\mathbb X}\times\widehat{\mathbb X}^*$ as follows:

$$ \hat{e}_n=(e_n,0) \quad \mbox{and}\quad \hat{e}^*_n(x,\lambda x_0)=e^*_n(x)+n\lambda.$$

Then $\{\hat{e}_n\}_{n=1}^\infty$ is a basic sequence in $\widehat{\mathbb X}$ (in fact a Schauder basis of $\mathbb{X}\oplus\{0\}$). However, the associated projection operators $\widehat{S}_N$ are not uniformly bounded, since

$$\widehat{S}_N(0,x_0)=\sum_{n=1}^N\hat{e}^*_n(0,x_0)e_n=\sum_{n=1}^Nne_n,$$ and therefore $$\|\hat{S}_N-\hat{S}_{N-1}\|\geq N.$$

In this example the dual functionals $\hat{e}^*_n$ have unbounded norms in ${\widehat{\mathbb X}^*}$ , but one could construct an example with bounded dual functionals, by setting $$ \hat{e}^*_n(x,\lambda x_0)=e^*_n(x)+\lambda,$$ provided that $$\|\widehat{S}_N(0,x_0)\|_{\widehat{\mathbb X}}=\|\sum_{n=1}^Ne_n\|_{\mathbb X}\to\infty,\quad \mbox{as }N\to\infty,$$ which happens, e.g. when $\mathbb{X}$ is super-reflexive.

One could still ask whether, for normalized biorthogonal systems $\{e_n,e^*_n\}_{n=1}^\infty\subset\mathbb{X}\times\mathbb{X}^*$ in spaces $\mathbb X$ such as $\ell_\infty$ or $L_\infty$, the equivalence in ($*$) may be true.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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