Abstract:
The restricted product over $X$ of copies of the $p$-adic numbers $\mathbb{Q}_p$, denoted $\mathbb{Q}_p(X)$, is self-dual and is the natural $p$-adic analogue of Hilbert space. The additive group of this space is locally compact and the continuous endomorphisms of the group are precisely the continuous linear operators on $\mathbb{Q}_p(X)$.
Attempts to develop a spectral theory for continuous linear operators on $\mathbb{Q}_p(X)$ will be described at an elementary level. The Berkovich spectral theory over non-Archimedean fields will be summarised and the spectrum of the linear operator $T$ compared with the scale of $T$ as an endomorphism of $(\mathbb{Q}_p(X),+)$.
The original motivation for this work, which is joint with Andreas Thom (Leipzig), will also be briefly discussed. A certain result that holds for representations of any group on a Hilbert space, proved by operator theoretic methods, can only be proved for representations of sofic groups on $\mathbb{Q}_p(X)$ and it is thought that the difficulty might lie with the lack of understanding of linear operators on $\mathbb{Q}_p(X)$ rather than with non-sofic groups.