Consider a family of maps $f_p\colon \ell^p(\mathbb Z,\mathbb R)\to \ell^p(\mathbb Z,\mathbb R)$, $p\ge 2$, where $$ f_q = f_p\big|_{\ell^q},\qquad \forall\; 2\le q\le p. $$ Moreover,
- $f_p\colon \ell^p\to \ell^p$ is real analytic for any $p\ge 2$,
- $f_p\colon \ell^p\to \ell^p$ is one-to-one for any $p\ge 2$,
- $f_p\colon \ell^p\to \ell^p$ is a local diffeomorphism everywhere. More to the point, $d_xf_p\colon \ell^p\to \ell^p$ is a compact perturbation of the identity at any point $x\in\ell^p$.
Therefore, $f_p$, $p\ge2$, is a Fredholm map, that is its differential is a Fredholm operator at each point. However, we do not know whether $f_p$ is proper, i.e. whether preimages of compact sets are compact.
Furthermore, for $p=2$ we have the additional properties
- $f_2$ is onto and hence is a diffeomorphism $f_2\colon \ell^2\to \ell^2$,
- $\|f_2(x)\|_2 = \|x\|_2$.
Is $f_p$ onto also for $p > 2$?
Since $f_p\big|_{\ell^2} = f_2$ the range of $f_p$ contains $\ell^2$ and hence is dense in $\ell^p$ for any $p\ge 2$. The range is also open since $f_p$ is a local diffeomorphism. One may thus equivalently ask:
Is the range of $f_p$ closed for $p > 2$?
Any suggestions how to attack these kind of problems?
