some questions about the Robba ring Notations and definitions
Let $p$ be a prime integer, $k$ be a perfect field of characteristic $p$ and  $W(k)$ its ring of Witt vectors.
Definition 1
We put  $$ \mathcal{R}_r=\bigg\{  \sum_{i\in \mathbf{Z}}a_iu^i: a_i\in W(k)[1/p],  \lim_{i\to \pm \infty} |a_i|\rho^i=0, \rho\in  {[} p^{-r}, 1{)}   \bigg\} $$
In other words, elements of $\mathcal{R}_{r}$ are Laurent series $ \sum_{i\in \mathbf{Z}}a_iu^i$ that satisfies $|a_i|\rho^i\to 0$ when $i\to +\infty$ for any $0<\rho<1$, and $|a_i|\rho^i\to 0$ when  $i\to -\infty$ for any $p^{-r} \leq \rho < 1$.
We define Robba ring to be $$ \mathcal{R}=\bigcup_{r>0}\mathcal{R}_r.  $$
In other words, elements of $\mathcal{R}$ are Laurent series $ \sum_{i\in \mathbf{Z}}a_iu^i$ that satisfies $|a_i|\rho^i\to 0$ when $i\to +\infty$ for any $0<\rho<1$, and there exists some $r>0$ such that $|a_i|\rho^i\to 0$ when  $i\to -\infty$ for any $p^{-r} \leq \rho < 1$.
Definition 2
For any $0< \rho < 1$, we define the $r$-Gauss norm over $\mathcal{R}$ as follows:
$$  \bigg|   \sum_{i\in \mathbf{Z}}a_iu^i   \bigg|_{\rho}=\sup_i\{ |a_i|\rho^i \}.  $$
Definition 3

*

*The ring $\mathcal{R}_r$ carries a Fréchet topology, in which a sequence converges if and only if it converges under the $\rho$-Gauss norm for all $\rho \in {[}p^{-r}, 1{)}$. (For this topology, $\mathcal{R}_{r}$ is complete.)

*The ring $\mathcal{R}$ carries a limit-of-Fréchet topology, or $LF$ topology. This topology is defined on $\mathcal{R}$ by taking the locally convex direct limit of the $\mathcal{R}_{r}$ (each equipped with the Fréchet topology). In particular, a sequence converges in $\mathcal{R}$ if it is a convergent sequence in $\mathcal{R}_r$ for some $r>0$.

Notations:
Let $E(u)\in W(k)[u]$ be the Eisenstein polynomial of $\pi.$ $$\lambda:=\prod_{n=0}^{\infty}\varphi^n(E(u)/E(0))\in \mathcal{R}$$
(Recall that $\frac{E(u)}{E(0)}$ is of the form $1+a_1u+a_2u^2+\cdots +\frac{u^e}{p\cdot unit}$ with $v_p(a_i)\geq 0$ and we can write uniquely $\lambda=\sum_{i\geq 0}\lambda_i u^i$ with $\lambda_i\in W(k)$.)
Put $\mathcal{R}^{+}$ for the series of $\mathcal{R}$ with nonnegative powers of $u$ and $\mathcal{R}^{-}$ for the series with negative powers.
The reason I care about $\lambda$ is because I care about the operator $-\lambda u\frac{d}{du}$ over the Robba ring, usually noted $N_{\nabla}$ in literature: for example in Kisin's article "Crystalline representations and F-crystals." )

Questions 1. (I gave a negative answer below)
In brief, I want to know how big is the image of the operator $-\lambda u \frac{d}{du}$ over the Robba ring modulo
$\mathcal{R}^{+}$: can any element $x\in\mathcal{R}^{-}$ be written in
the form $x=-\lambda u\frac{d}{du}(x_1)+x_2$ with $x_1\in \mathcal{R}$
and $x_2\in \mathcal{R}^{+}$? If not, a counter example?

Remark 1
The difficulty to describe $Im(-\lambda u \frac{d}{du})$ lies in the factor $\lambda$: without $\lambda$, $-u\frac{d}{du}$ is very well-behaved. In other words, multiplication by $\lambda$ is mysterious for me.

The following questions are what I expect to help approaching an answer. Any remarks or references for any of the big or small questions below are welcomed.

1.opens
For me, Robba ring $\mathcal{R}$ is more complicated than a metric space: you have to deal with a series of $r$-Gauss norm where $r$ takes values in an interval.

*

*Is there a reasonable definition of "(fundamental system of) open neighborhoods of $0$" in $\mathcal{R}$?

Reasonablely, a series is close to $0$ when it is so under all $r$-Gauss-norm where $r$ takes value in some interval. By the fact $\mathcal{R}_{r}$ is complete for the Fréchet topology, it is a good candidate of closed neighborhood of zero, and it gives a system when $r$ changes: $r_1>r_2$ implies that $R_{r_1}\subset R_{r_2}$.
1.1 What should a continuous map over the Robba ring look like?

2.radius
For a given element $x=\sum_{n\in\mathbf{Z}}a_nu^n\in \mathcal{R}$, there exists a smallest $0<r<1$ such that for all $\rho\in (r, 1), \sum_{n\in\mathbf{Z}}a_n\rho^{n}$ converges (i.e. $a_n\rho^{n}\to 0$ when $n\to +\infty$ or $n\to -\infty$). I want to define $r$ the radius of $x$. (Is this well-defined?) Now I want to study what operations can influence the radius of an element. For example: Frobenius map ($\varphi: \sum_{n\in\mathbf{Z}}a_nu^n\mapsto \sum_{n\in\mathbf{Z}}\varphi(a_n)u^{np}$) and its inverse map $\psi$ (when it is well-defined) obviously changes the radius. How about other operations, like multiplication by an element?


*If $x$ has radius $r_1$ and $y$ has radius $r_2$, can we have a formula for the radius of $xy$? Seems pessimistic as for example take any monimal $u^N$ for $N\in \mathbf{N}$, having "radius 0" by our definition, but $u^N \cdot \sum_{n\in\mathbf{Z}}a_nu^n$ doesn't change the radius no matter how big $N$ is. (You really have to be able to change the power of $u^n$ for $n\gg 0$ to change the radius of $x=\sum_{n\in\mathbf{Z}}a_nu^n$.) So:

2.1 (Less related) Are there results about how
$\mathcal{R}_{r_1}\cdot \mathcal{R}_{r_2}\subset \mathcal{R}_{r_3}$ with a formula $r_3=f(r_1, r_2)$? (At least, $\mathcal{R}_{r_1}\cdot \mathcal{R}_{r_2}\subset \mathcal{R}_{r_2}$ when $\mathcal{R}_{r_1}\subset \mathcal{R}_{r_2}$, since it is a ring. )
2.2 How is $\lambda \cdot x$ changing the radius of $x$?

3.image of multiplication by $\lambda$ modulo $\mathcal{R}^{+}$
About the image of multiplication by $\lambda$ (hence more or less $-\lambda u \frac{d}{du}$) over the Robba ring:
2.2.1 (Main question) Can you determine what are the elements of $\mathcal{R}$ that can be written of the form $\lambda \cdot x$ for some $x\in\mathcal{R}$?
2.2.2 Is the multiplication by an element, for example $\lambda$ a continuous map? (Remark that for any fixed $r$-Gauss norm it maps Cauchy sequence to Cauchy sequence.)
2.2.3 Does $Im(\cdot \lambda)$ modulo $\mathcal{R}^{+}$ contains an open neighborhood of $0$?
Remark 2
By some computations, $Im(-\lambda u \frac{d}{du})$ contains $\mathcal{R}_{r}$ modulo $\mathcal{R}^{+}$ for some $r$ close to $1$ and also I see that any finite sum $\sum_{finite}a_nu^n$ is inside $Im(-\lambda u \frac{d}{du})$ modulo $\mathcal{R}^{+}$. This implies, by a little more computation, that the image is dense inside $\mathcal{R}$ modulo $\mathcal{R}^{+}$. Hence I expect also some approaches from functional analysis to tell me how big it is.

As I observed my previous question has a negative answer, I now change the question as follows:

Question 2:
Can any element $x\in\mathcal{R}^{-}$ of the form $x=\sum_{i>0, p\nmid
> i}a_{-i}u^{-i}$ be written in the form $x=-\lambda
> u\frac{d}{du}(x_1)+x_2$ with $x_1\in \mathcal{R}$ and $x_2\in
> \mathcal{R}^{+}$?


Question 3:
https://mathoverflow.net/questions/390160/a-question-on-the-robba-ring


 A: The question posted was:
Can any element $x\in\mathcal{R}^{-}$ be written in the form $x=-\lambda u\frac{d}{du}(x_1)+x_2$ with $x_1\in \mathcal{R}$ and $x_2\in \mathcal{R}^{+}$.
I think the answer of my question is negative by the following arguments (omitting some computations).
Proof:
For any $x=\sum_{n>0}a_{-n}u^{-n}\in \mathcal{R}$, suppose there exist elements $y=\sum_{i\in \mathbf{Z}}b_iu^i$ and $z=\sum_{j\geq 0}c_ju^j$ in $\mathcal{R}$ such that
$$ x=N_{\nabla}(y)+z. $$
Take the part inside $\mathcal{R}^{-}$,  we have
$$ x=N_{\nabla}(y^{-})$$
with $y^{-}=\sum_{i<0}b_iu^i$. More precisely, we have
$$ \sum_{n>0}a_{-n}u^{-n}=\sum_{n<0}\big(\sum_{i\geq 0}\lambda_i (-n-i)b_{-n-i} \big)u^{-n}. $$
This implies that for any $n>0,\ a_{-n}=\sum_{i\geq 0}\lambda_i (-n-i)b_{-n-i}$ . We claim that for any $i>0$, we have
$$ b_{-i}=\frac{1}{i}(a_{-i}+a_{-i-1}(-\lambda_1)+ a_{-i-2}(\lambda_1^2+\lambda_2)+a_{-i-3}(-\lambda_1^3+2\lambda_1\lambda_2-\lambda_3)+\cdots ), $$
in other words $b_{-i}=\frac{1}{i}\sum_{j\geq i}a_{-j}P_{j-i,j}$ (where $P_{j-i,j}$ is some homogeneous polynomial of degree $j-i$ with $\lambda_n$ having degree $n$). Indeed, this can either be verified directly or by the observation: that for any $j>0$ we have $u^{-j}=N_{\nabla}(\sum_{n=0}^{j-1}\frac{1}{j-n}P_{n,j}u^{n-j})+z$ with $z\in \mathcal{R}^{+}$, hence
$$ \sum_{n>0}a_{-n}u^{-n}=-\sum_{n>0}\big(\sum_{i\geq 0} \big(\sum_{j\geq i}a_{-j}P_{j-n-i,j})\lambda_i \big)u^{-n}=N_{\nabla}\big(\sum_{n>0}(\frac{1}{n}\sum_{j\geq n}a_{-j}P_{j-n,j})u^{-n}). $$
But by some estimation of $|P_{i,j}|$, $b_{-i}=\frac{1}{i}\sum_{j\geq i}a_{-j}P_{j-i,j}$ is not well-defined when $x=\sum_{n>0}a_{-n}u^{-n}$ is outside some subring $\mathcal{R}_{r}\subsetneq \mathcal{R}$ ($r$ is determined by $E(u)$).
This tells that  for certain $E(u)$, there exists element $x\in\mathcal{R}^{-} \cap (\mathcal{R}\setminus\mathcal{R}_{r})$ ($r$ is determined by $E(u)$) that can not be written in the form $x=N_{\nabla}(x_1)+x_2$ with $x_1\in \mathcal{R}$ and $x_2\in \mathcal{R}^{+}$.

With the same proof, I think the Question 2 posted is also negative.
