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

Hervé, Loïc

Ann. Inst. Henri Poincaré, Probab. Stat. 30, No. 3, 437-466 (1994).

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

Keywords:

family of non-negative measures; positive operator; transfer operators