Subordinate semigroups and order properties

Abstract

AbstractLetSt= exp{−tH},Tt= exp{−tK}, beC0-semigroups on a Banach space. For appropriatefone can define subordinate semigroupsSft= exp{−tf(H)},Ttf= exp{−tf(K)}, onand examine order properties of the pairsS, T, andSf,Tf. If, =Lp(X;dv)we defineSt≽Tt≽ 0 ifSt−TtandTtmap non-negative functions into non-negative functions. Then forpfixed in the range 1 >p> ∞ we characterize the functions for whichSt≽Tt≽ 0 impliesSft≽Tft≽ 0 for eachLp(X;dv)and the converse is true for allLp(X;dv). Further we give irreducibility criteria for the strict ordering of holomorphic semigroups. This extends earlier results forL2-spaces.