Extreme doubly stochastic operators and characteristic functions
Let the measure space (X,m) be the unit interval [0,1] with Lebesgue measure. A linear operator T on \(L^ 1(X,m)\) is doubly stochastic if Tf\(\geq 0\) for \(f\geq 0\), \(T1=1\), and \(\int Tf dm=\int f dm\). The doubly stochastic operators form a convex set, and the extreme points of this set are called extreme doubly stochastic operators. The authors are concerned with the problem: ''Determine whether or not every extreme doubly stochastic operator maps some nontrivial characteristic function to a characteristic function.'' The idea behind this problem seems to data back to the work of J. V. Ryff in the 1960's, and a recent paper of \textit{J. V. Ryff} [J. Reine Angew. Math. 302, 116-136 (1978)] was an important contribution to this general area of mathematics. The authors solve the problem by constructing an extreme doubly stochastic operator T such that if \(T\chi_ A=\chi_ B\), then \(m(A)=0\) or 1.
scientific article; zbMATH DE number 3925628
scientific article