The Borel complexity of non-Archimedean operator ranges

Abstract

We completely classify the possible complexity classes of non-closed operator ranges on separable Banach spaces over a Polish non-Archimedean non-trivially valued field. These are precisely ${\mathbf{\Pi }}_{1+\lambda +n+2}^0$ for a countable ordinal $\lambda$ that is either zero or a limit ordinal, and a finite ordinal $n$. Considering Fréchet spaces produces the additional complexity classes ${\mathbf{\Pi }}_{\lambda }^0$ for a countable limit ordinal $\lambda$.

12 pages