## Quasi-compact positive operators. Applications to transfer operators. (Étude d’opérateurs quasi-compacts positifs. Applications aux opérateurs de transfert.)(French)Zbl 0804.47038

Summary: Let $$X$$ be a compact metric space, $$\{\eta(x,\cdot)$$, $$x\in X\}$$ a family of non-negative measures on $$X$$, and $$P$$ the positive operator defined by $Pf(x)= \int_ X f(y)\eta(x,dy),$ which we suppose to be bounded on $${\mathcal C}(X)$$ and quasi-compact on the space $$E^ \alpha$$ of $$\alpha$$- Hölderian functions, where $$0< \alpha\leq 1$$. We know that there exists an integer $$\nu$$ such that $\bigcup^{+\infty}_{l=1} \text{Ker}(P'- \rho)^ l= \text{Ker}(P'- \rho)^ \nu,$ where $$\rho$$ is the spectral radius of $$P$$, and $$P'= P_{| E^ \alpha}$$. The object of this work is to give a necessary and sufficient condition for the existence of a $$\gamma>0$$ in $${\mathcal C}(X)$$ such that $$(P- \rho)^ \nu\gamma= 0$$. In this case we describe, when $$\nu=1$$, the space $$\text{Ker}(P- \rho)$$, and for $$\rho=1$$ and $$\nu\geq 1$$ the $$P$$-invariant measures. We give applications to transfer operators.

### MSC:

 47B65 Positive linear operators and order-bounded operators 47B07 Linear operators defined by compactness properties 47B38 Linear operators on function spaces (general) 37A99 Ergodic theory
Full Text: