sum of Perron-Frobenius operators - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:50:16Z http://mathoverflow.net/feeds/question/43875 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43875/sum-of-perron-frobenius-operators sum of Perron-Frobenius operators filiz 2010-10-27T21:10:38Z 2010-11-25T08:18:42Z <p>My operator is the transfer operator $P$ on $L^1$ functions defined on compact $X$. It is the pre-dual of the operator $U:L^∞ \rightarrow L^∞$ defined by $U(ϕ)=ϕ\circ f$, for a fixed map f on X. I have PU=Id and UP is the projection. Now my specific question is, if $P_1$(h)=h and $P_2$(g)=g for g,h ∈ L1, and if 1 is the leading simple isolated eigenvalue for both $P_1$ and $P_2$, then does $(P_1+P_2)/2$ have 1 as a leading eigenvalue and what about the corresponding eigenfunction? </p> http://mathoverflow.net/questions/43875/sum-of-perron-frobenius-operators/47310#47310 Answer by Anthony Quas for sum of Perron-Frobenius operators Anthony Quas 2010-11-25T08:18:42Z 2010-11-25T08:18:42Z <p>Not clear what you mean by this. If $P$ has a non-trivial kernel, (say $Pf=0$) then isn't $f+aUf+a^2U^2f+\ldots$ an $L^1$ function and an eigenfunction of $P$ with eigenvalue $a$?</p> <p>I think that to have isolated eigenvalues you need to be working in a smaller Banach space.</p>