Positivity and monotonicity properties ofC 0-semigroups. I

If exp {?tH}, exp {?tK}, are self-adjoint, positivity preserving, contraction semigroups on a Hilbert space ?=L ²(X;dμ) we write (*) $$e^{ - tH} succcurlyeq e^{ - tK} succcurlyeq 0$$ whenever exp {?tH}-exp {?tK} is positivity preserving for allt≧0 and then we characterize the class of positive functions for which (*) always implies $$e^{ - tf(H)} succcurlyeq e^{ - tf(K)} succcurlyeq 0.$$ This class consists of thef∈C ^∞(0, ∞) with $$( - 1)^n f^{(n + 1)} (x) geqq 0,x in (0,infty ),n = 0,1,2, ldots .$$ In particular it contains the class of monotone operator functions. Furthermore if exp {?tH} isL ^p (X;dμ) contractive for allp∈[1, ∞] and allt>0 (or, equivalently, forp=∞ andt>0) then exp {?tf(H)} has the same property. Various applications to monotonicity properties of Green's functions are given.