Perhaps my question is naive, but let me try.


Take a (real or complex) vector space $V$ and consider an ideal $\mathcal{I}$ of subsets of $V$ with the following property (call it (\*)): for each linear map $A\colon V\to V$ and each $S\in \mathcal{I}$ we have $A(S)\in \mathcal{I}$.


The ideal of finite sets enjoys this property. Is there a maximal ideal (that is, such that the set $\{E\setminus S\colon S\in \mathcal{I}\}$ is an ultrafilter) with (\*)?


What if we replace vector spaces and linear maps by Banach spaces and bounded linear operators?



Ultrafilters over vector spaces
Every ideal satisfying $(\\*)$ is included in an ideal maximal with respect to the property of satisfying $(\\*)$, by Zorn's lemma. However, such an ideal need not be maximal as an ideal.


No maximal ideal can satisfy $(\*)$, apart from trivial cases ($0$-dimensional vector spaces). Let $X$ be a subset of the base field $F$ such that $F$ is the disjoint union of $X$, $-X$, and $\{0\}$ (e.g., $X=(0,\infty)$ if $F=\mathbb R$). We can choose a basis of the vector space, hence we may assume that $V$ is the space of almost everywhere $0$ functions $I\to F$. Fix $i\_0\in I$, let $A$ be the linear function which negates the $i\_0$th coordinate, $B$ the function which sets the $i\_0$th coordinate to $0$, and $S=\{v\in V:v(i\_0)\in X\}$. If $I$ is a maximal ideal, either $S$ or its complement belongs to $I$. However, $V=S\cup A(S)\cup B(S)$ and similarly for $V\smallsetminus S$, hence $V\in I$, a contradiction. The same argument works for vector spaces over any field of characteristic other than $2$.


