The Lomonosov type theorems and the invariant subspace problem for non-archimedean Banach spaces

In this paper, we study the existence of invariant (and even hyperinvariant) subspaces of bounded operators on a non-archimedean Banach space $E=(E,\Vert .\Vert )$ over a valued field $K$ equipped with a non-trivial non-archimedean valuation $|.|$. Specifically, we consider compact operators and operators that commute with a compact operator. First we show that if E has a base, then any compact operator T such that ${\lim }_n{\Vert T^n\Vert }^{\frac{1}{n}}>0$ has a finite-dimensional hyperinvariant subspace. Next we show that if $K$ is locally compact, then every compact operator T on E has a hyperinvariant subspace. Afterward, assuming that $K$ is spherically complete or E is of countable type, we provide a necessary condition for a bounded operator on E to have a hyperinvariant subspace. We demonstrate that the classical Lomonosov Invariant Subspace theorem does not hold in the case where $K$ is non-spherically complete. Finally, we prove Lomonosov type theorem for spectral quasinilpotent operators, when $K$ is locally compact.