Vol. 40, No. 1, 1972

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Extreme Markov operators and the orbits of Ryff

Ray C. Shiflett

Vol. 40 (1972), No. 1, 201–206

Let X be the unit interval with the Lebesgue structure and let m be Lebesgue measure. A Markov operator with invariant measure m is an operator T on L(X,m) such that T1 = 1 and Tfdm = fdm for all f in L(X,m). If 𝜃 is a measure-preserving transformation on X, then 𝜃f = f 𝜃 defines a Markov operator. Each such 𝜃 is an extreme point in the convex set of Markov operators.

Let Ω(f) be the set of all g L1(X,m) such that Tf = g for some Markov operator T. This convex set is called the orbit of f. The extreme points of Ω(f) are equimeasurable to f and arise from Markov operators of the form 𝜃σ. This paper shows the connection between extreme points of the set of Markov operators and the extreme points of Ω(f). The set of Markov operators which carry f to a given extreme point of Ω(f) is shown to contain an extreme Markov operator. The Markov operators of the from 𝜃σ are shown to be extreme when 𝜃 is invertible. It is also shown that not all extreme operators factor into 𝜃σ and that there are 𝜃 and σ such that 𝜃σ is not extreme.

Mathematical Subject Classification
Primary: 28A65
Received: 30 April 1970
Published: 1 January 1972
Ray C. Shiflett