Let B be either the space L 2 with respect to a finite measure or else the continuous functions vanishing at infinity for some locally compact separable metric space S. In the former case, we say  ...

Let B be either the space L₂
with respect to a finite measure or else the continuous functions vanishing at infinity for some locally compact separable metric space S. In the former case, we say

  for almost every x, in the latter case if
  for all x. A semi group is non-negative if
 implies
  for all t ≥ 0. Suppose that P_t
is a semi group, the space B contains the constant functions, and Pt1 = 1 for all t. Show that P_t
is a contraction if and only if P_t
is non-negative.