I have little experience with functional analysis beyond an undergraduate basic course, and I'm dealing with the following problem:
let $V$ be an infinite-dimensional locally convex (but not normed!) vector space, and let $O:V\rightarrow V$ be an invertible continuous operator; how can I decide whether it has eigenvalues or not?
The problem, sits in a larger framework: let $G$ be a topological group of operators as above; is there a common eigenvector for all these operators?
I do not know what are the tools for this type of problems in an infinite-dimensional setting. The only related theorem that I know proves that the spectrum of an element in a Banach algebra is nonempty (the proof that I know is based upon Liouville's theorem from complex analysis and clearly cannot be mimicked here); I do not know of any result regarding the eigenvalues and not the spectrum (except for the classical result in a finite-dimensional setting). The fact that I do not have a norm can only complicate things, I believe. Any suggestion or bibliographical hint would be appreciated. Thank you.
(Somebody already registered please add the tag "eigenvalue", I'm not allowed to.)