Common eigenvector - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T06:29:36Z http://mathoverflow.net/feeds/question/105906 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105906/common-eigenvector Common eigenvector Alex M. 2012-08-30T07:01:10Z 2012-08-30T11:19:08Z <p>I have little experience with functional analysis beyond an undergraduate basic course, and I'm dealing with the following problem:</p> <p>let $V$ be an <b>infinite</b>-dimensional locally convex (but not normed!) vector space, and let $O:V\rightarrow V$ be an <b>invertible</b> continuous operator; how can I decide whether it has eigenvalues or not?</p> <p>The problem, sits in a larger framework: let $G$ be a topological group of operators as above; is there a <b>common</b> eigenvector for all these operators?</p> <p>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.</p> <p>(Somebody already registered please add the tag "eigenvalue", I'm not allowed to.)</p>