FUNCTIONAL POSITIVITY AND INVARIANT SUBSPACES OF SEMIGROUPS OF OPERATORS

Let chi be a topological space, and with its Borel structure, a standard Borel space, and m a sigma-fulite regular Borel measure on chi such that L2(chi, m) is of dimension at least two. An operator on L2(chi, m) is called (functionally) positive if it maps non-negative fttnctions to non-negative functions. Suppose S is a semigroup of positive integral operators on L2(chi, m). We show that S has a non-trivial invariant subspace if every operator in S is quasinilpotent. We prove the existence of special kinds of bases of the ranges of positive integral idempotent operators which consist of only non-negative functions. Using these bases, we prove that S has a non-trivial invariant subspace if it contains a non-zero compact operator and r(AB) less-than-or-equal-to r(A)r(B) for all A, B in S.