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,‖.‖) 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 limn⁡‖Tn‖1n>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.