of bounded real sequences is denoted by l , with norm || = sup |x|. Note that fÎl, with q < Þ |x| 0, and that l Ì

Corollary 2.5: If T S is induced by P, then Tt is a positive linear operator mapping l l such that T t g = g P t = Pg " g l and |Tt|

Proof: Given g l , S(Ttg)f = Sg(Tf) = Sg(f P) = Sg(Ptf) = S(gPt)f " f l1Þ Ttg=TtgPt = P g " gl .If g=á||ynñ, then||Ttg|| =supi{|Sjpijyj|}

Theorem 2.6: Let T be a positive bounded linear operator on l1. T Î S iff T t l = 1.