SEMIGROUPS. D. Pfeifer. Communicated by Jerome A. Goldstein Dedicated to E.S. Lyapin on his 70th Birthday

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1 A NOTE ON ~ROBABILISTIC REPRESENTATIONS OF OPERATOR SEMIGROUPS D. Pfeifer Commuicated by Jerome A. Goldstei Dedicated to E.S. Lyapi o his 70th Birthday I the theory of strogly cotiuous semigroups of bouded liear operators {T(t.) t > O} o a Baach space 9: may represetatio theorems of the form (1) T(t;)f lim ' ~(T(~) )f (2) T(t;) f lim 'lit; (R () )f (3) T( t;)f 1 lim 'lit; (I +~A)f (t; > 0, f 89:) have bee established by several authors ([2J, [3], [5J, [7], [8J, [9] ), where 'lit; is a sui table f:.1ctio aalytic i some iterval 00 _ At [0,8J with 8 > 1, R(A) = f e T(t)dt for sufficietly large A deo otes the resolvet of the semigroup, I deotes the idetity operator ad A deotes u le correspodig ifiitesimal geera tor. (Note that (3) is oly meaigful if A is bouded.) The commo backgroud all of these represetatio theorems is probabilistic i that 'lit; is the geeratig fuctio of a o-egative iteger-valued radom variable N (see [4J for defiitios) with expec tatio E(N) = U [3], [7], [8]) ; i fact, relatios (1), (2) ad (3) are i some sese a cosequece of the famous law of large umbers i probability theory. 1

2 The aim of this ote ow is to prove that uder mild coditios oly these probabilistic represetatios are possible. THEOREM 1. Let ~ be aalytic i some iterval [O, oj, > 1, with I:; o-egative coefficiets. The if ay of the three represetatios give by (1), (2) or (3) holds for a arbitrary strogly cotiuous o-periodic semigroup {T(t) I t ~ O} with IIT(t)11 > 0, ~I:; is ecessarily the geeratig fuctio of a o- egative iteger- valued radom variable N with expectatio E(N) = 1:;. I this case, the represetatios (1) ad (2) hold true for every strogly cotiuous semigroup, a d (3) holds true i case that A is bouded. PROOF. Let ~I:;(t) = L ak( l:;) t k for ~ t ~ 00 k=o The ~~(1) = I a ( 1:;) > sice otherwise ~!:; := 0, hece IIT(!:;) II = c, k=o k a k (!:;) which is a cotradictio. But the by {~ t;( I)} a probability distrib~ tio of some o-egative itege r - valued radom variable N is give, with beig its geeratig fuctio. By the iequality x < l. e XY -y grable with for arbitrary x _> 0, y > 0, N is ite E(N) < _1_ E(e N lo) - l < Also, sice ~I:; ad hece ~~ are aalytic i [o,oj with > 1, the characteristic fuctio of N is aalytic i some complex eighbourhood of the origi, hece all of the relatios (4) * T (1;)f lim -- {~~ (T* (~) ) }f (5) * T (1;) f * * lim {f I:; (R ())} f -- 2

3 (6) * T (1;)f * 1 * lim {' I;(I +~A )} f, f ef!{ * hold true for every strogly cotiuous semigroup {T (t) I t ~ o} (the latter oly, if A* is bouded) ([7], Corollary 2; [8], (6)). 1 1 Let S deote oe of the operators T(~), R() or I +~A, correspodig to which of the relatios (1), (2) or (3) holds. Choose f e f!{ such that IIT(t;)fll > 0, the 0< IIT(I;)fll = limll{'!'i;(s)}fll lim('!'i;(l) )ll {'!'~(S) }fll with lim II {'!'~(S) }fll IIT( I; )f11 < '" by (4), (5) ad (6), implyig '!'I; (1) ~ 1. Let further g e f!{ be such that II T (1;) gil > o. The lim with limll {'!'~ (S) }gll = IIT(1;)gll > 0 by (4), (5) ad (6), implyig '!'1;(1) ::. 1. That is, '!' 1; (1) = 1 ad hece '!'I; = '!'i;" * Agai by (4), (5) ad (6), T(1;) f T ( 1;) f for all f e P1 by assumptio, hece T(r,) T( I; ). Suppose ow 1; f 1;, say 1; < 1;. The for I; - r" T(I;)T() T(1; + ) T(I;), hece the semigroup is periodic with period which is a cotradictio to the assumptio. Hece 1; = 1;, ad the theorem is proved. Note that the a b ove theorem is a strog extesio of the geeral covergece theorem i [lj i the case of strogly cotiuous semigroups sice oly oe (essetially) arbitrary "test fuctio" is eeded. 3

4 The assumptio of o-periodicity i the above theorem is oly ecessary to guaratee that E(N) = ~. Without this assumptio, the theorem e ssetially remais valid i that the represetatios (1), (2) ad (3) hold true i geeral if T(~)f is r eplaced by T(s)f. Of course, the poit i the above theore m is the assumptio that the coefficiets o ' ~ are o-egative. Oe could ask ow whether geerally oly probabilistic represetatios of the form (1), (2) or (3) are possible. But t~is is ot true as ca be see by the followig o-probabilistic exte sio of Kedall's formula [5J. THEOREM 2. Let ' ~(t) = l-~+~t for ~ > 1. The (1), (2) ad (3) hold true for every uiformly cotiuous semigroup {T(t) It> ol PROOF. Let agai S deote T(!), R() V (S -I). The A = lim V, ad 1 or I +-A, ad let -l I k=o ~llv il e 1 3~11 V il < -e - 1 < -e - for sufficietly large. Sice by the expoetial formula, T(~) proved. ~V lim e the theorem is REMARK. I [7] it was implicitly assumed that the mappig t + T(t) is-measurable ad separably valued, which e.g. is true i the case of uiformly cotiuous semigroups. But the results i [6] remai also valid i the geeral strogly cotiuous case sice for a 4

5 suitable o-egative radom variable X {E (T (X)) }f E (T (X) f), f f defies a bouded liear operator o f ito f with the properties IIE(T(X))II ::.. E(IIT(X)II) ad E(T(X+Y)) = E(T(X))oE(T(Y)) for idepedet radom variables X ad Y, where E (T (X) f) for f f is to be uderstood as a Pettis expectatio i the sese of Mourier [6J (which exists sice the mappigs t ~ T(t)f are cotiuous, hece measurable ad separably valued (see also [2])). REFERENCES [1] Butzer, P.L. ad Beres, H.: Semi- Groups of Operators ad Approximatio, Spriger-Verlag, Berli (1967). [2] Butzer, P.L. ad Hah, L.: A probabilistic approach to represetatio formulae for semigroups of operators with rates of covergece. Semigroup Forum 21 (1980), [3] Chug, K.L.: O the expoetial formulas of semi-group theory. Math. Scad. 10 (1962), [4J Feller, W.: A Itroductio to Probability Theory ad Its Applicatios. Vol. I. No.4. Wiley, New York (1968). [5] Kedall, D.G.: Berstei polyomials ad semigroups of operators. Math. Scad. 2 (1954), [6] Mourier, E.: Elemets aleatoires das u espace de Baach. A. Ist. Heri Poicare 13 (1953), [7] Pfeifer, D.: O a geeral probabilistic represetatio formula for semigroups of operators. J. Math. Res. Exp. 2 (1982),

6 [81 Pfeifer, D.: O a probabilistic represetatio theorem of operator semigroups with bouded geerator. J. Math. Res. Exp. 4 (1984). No.1. 19] Shaw, S. Y.: Approximatio of ubouded fuctios ad applicatioyb to represetatios of semigroups. J. Approx. Theory 28 (1980), Istitut fur Statistik ud Wirtschaftsmathematik Techical Uiversity Aache Wu1lerstr Aache West-Germay 6