Extreme invariant positive operators on $L_{p}$-spaces

Abstract:

Let $(X,\mathfrak {A},\mu )$ and $(Y,\mathfrak {B},\nu )$ be finite positive measure spaces. In this note we present characterizations of the extreme points of the convex set of all positive linear operators $T:{L_p}(\mu ) \to {L_q}(\nu )$ with $T{{\mathbf {1}}_X} = {{\mathbf {1}}_Y}$ which are invariant with respect to a semigroup of positive constant preserving contractions on ${L_p}(\mu ),1 \leqslant p < \infty ,1 \leqslant q \leqslant \infty$.