My operator is the transfer operator $P$ on $L^1$ functions defined on compact $X$. It is the predual 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?

Not clear what you mean by this. If $P$ has a nontrivial kernel, (say $Pf=0$) then isn't $f+aUf+a^2U^2f+\ldots$ an $L^1$ function and an eigenfunction of $P$ with eigenvalue $a$? I think that to have isolated eigenvalues you need to be working in a smaller Banach space. 

