Positive Linear Operators on LP and the Doeblin Condition

Abstract

This chapter focuses on positive linear operators on L p and the Doeblin condition. A classical result in the theory of Markov processes, by Doeblin, that can be stated in operator theoretical terms is discussed in the chapter. Doeblin's proof is of probabilistic nature, and a different proof is given by Yosida and Kakutani. Under a corresponding condition, Doeblin's result holds for a very large class of Banach lattices that in addition to all AM-spaces contains all L p -spaces, 1 < p ≤ ∞. If E = L p , 1 < p < ∞, is infinite dimensional, then E does not have an order unit nor is the composition of two weakly compact operators on E compact. Thus, a different approach is required. The chapter discusses the condition so that a positive linear operator T on Banach lattice E is said to satisfy a Doeblin condition.