A NOTE ON THE ALMOST EVERYWHERE CONVERGENCE OF ALTERNATING SEQUENCES WITH DUNFORD SCHWARTZ OPERATORS
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1 C O L L O Q U I U M M A T H E M A T I C U M VOL. LXII 1991 FASC. I A OTE O THE ALMOST EVERYWHERE COVERGECE OF ALTERATIG SEQUECES WITH DUFORD SCHWARTZ OPERATORS BY RYOTARO S A T O (OKAYAMA) 1. Inroducion. Le L p, 1 p, be he usual Banach spaces of real or complex funcions on a σ-finie measure space (X, F, µ). By a Dunford Schwarz operaor we mean a linear operaor T which maps he linear space L 1 + L ino iself and is a conracion of L p ino L p for each 1 p (i.e. T f p f p for all f L p ), and saisfies T f = lim n T f n almos everywhere whenever (f n ) is a sequence in L, f = lim n f n almos everywhere and sup n f n <. The following is known (see e.g. [9], [10]): If T is a linear conracion of L 1 ino L 1 and saisfies T f f for all f L 1 L, or if T is a linear operaor mapping 1<p< L p ino iself and is a conracion of L p ino L p for each 1 < p <, hen T can be uniquely exended o a Dunford Schwarz operaor. In his noe we deal wih a sequence (T n ) of Dunford Schwarz operaors on L 1 +L and discuss he almos everywhere convergence of he alernaing sequence T 1... T nt n... T 1 f (f L 1 + L ). Using an approximaion argumen involving maximal operaors and a resul of Akcoglu [1] which saes ha if f L p, l < p <, hen he alernaing sequence converges almos everywhere, we shall prove ha if f L 1 + L saisfies f log + ( f /a) dµ < for all a > 0, hen he alernaing sequence converges almos everywhere; hus a generalizaion of Akcoglu s resul will be obained. I should be noed here ha a similar resul has been announced in Assani [3]; bu we could no see he deails. (Afer he firs manuscrip of his paper was submied, he auhor could ge Assani s paper Roa s alernaing procedure wih non-posiive operaors (o appear in Adv. in Mah.), in which Assani deals wih Dunford Schwarz operaors defined on he real linear
2 98 R. S A T O space L 1 of a finie measure space. The auhor hinks ha Assani s paper does no include he resul of his noe.) 2. Resul Theorem. Le (T n ) be a sequence of Dunford Schwarz operaors on L 1 + L and le f L 1 + L be such ha f log + ( f /a) dµ < for all a > 0. Then lim n T 1... T nt n... T 1 f exiss a.e. on X. The heorem does no hold if f is only assumed o be in L 1 ; an example was given by Burkholder [4]. In case µ(x) =, i may happen ha here exiss a funcion f in L 1 + L which saisfies he condiion of he heorem bu is no in L 1 ; an example can be found in Fava [6]. As is easily seen, each f in L p, 1 < p <, saisfies he condiion of he heorem. P r o o f. I suffices o consider he case f 0. Given an ε > 0, pu e = f 1 {f ε} and g = f e where 1 A denoes he indicaor of a se A, and wrie f n = T1... TnT n... T 1 f (1) e n = T1... T nt n... T 1 e (n 1). g n = T1... TnT n... T 1 g I follows ha (2) f n = e n + g n and e n e ε (n 1). Since µ({g > 0}) = µ({f > ε}) <, we hen have g L 1 and furher g log + gdµ <. We now choose 0 < h L 1 wih 1 h min{g, 1}, and apply Doob s [5] and Sarr s [10] argumen as follows. Firs, le τ n denoe he linear modulus of T n (see e.g. [7], p. 159); hus τ n is a posiive Dunford Schwarz operaor on L 1 + L saisfying T n f τ n f for all f L 1 + L. By Lemma 2 in [10], seing g = g/h here exis finie measure spaces (X k, µ k ), k = 0, 1,..., for which X X k, X = X 0, µ 0 = h dµ, and posiive linear operaors S k from L 1 (X k 1, µ k 1 ) o L 1 (X k, µ k ) for which S k 1 = 1 a.e. (µ k ), S k 1 = 1 a.e. (µ k 1 ) and (3) S 1... S k[(τ k... τ 1 h)(s k... S 1 g)] = τ 1... τ k τ k... τ 1 g a.e. (µ 0 ). Since gh = g and log + g = log + g, i follows ha (4) g log+ g dµ 0 = g(log + g)h dµ = g log + g dµ <.
3 DUFORD SCHWARTZ OPERATORS 99 We nex choose a sequence (r ), = 1, 2,..., of funcions in L 2 such ha 0 r g a.e. on X, and wrie r = (g r )/h. From (3) and he fac ha 0 < h 1 i follows ha (5) S 1... S ks k... S 1 r τ 1... τ k τ k... τ 1 (g r ) a.e. (µ 0 ). Furher, from [5] or [10], if he usual probabiliy noaion is used, we may wrie (6) S 1... S ks k... S 1 r = E{E{ r (x 0 ) x k } x 0 } a.e. (P ), and (7) S k... S 1 r = E{ r (x 0 ) x k } = E{ r (x 0 ) x k, x k+1,...} a.e. (P ) where x k is he kh coordinae funcion on he produc space Ω = X 0 X 1... and P is he finie measure on Ω defined o make he x k sequence a Markov process wih iniial measure µ 0 = h dµ. Le M denoe he maximal operaor on L 1 (Ω, P ) defined by MX(ω) = sup E{X x k, x k+1,...}(ω) k 1 (ω Ω, X L 1 (Ω, P )). Then we have MX X for all X L (Ω, P ) and {MX>a} P ({MX > a}) 1 a {MX>a} X dp (a > 0, X L 1 (Ω, P )) (cf. e.g. [8], p. 69). Therefore Theorem 1 in [9] can be applied o infer ha here exiss a consan B > 0 such ha MX ( a dp B X log B X ) dp a a {B X >a} for all a > 0 and X R 1 (Ω, P ), where we le { R 1 (Ω, P ) = X L 1 (Ω, P ) : X log + X a } dp < for all a > 0. (I is known (cf. [6]) ha, since P is a finie measure, R 1 (Ω, P ) is a linear subspace of L 1 (Ω, P ), and X R 1 (Ω, P ) if and only if X log + X dp <.) On he oher hand, since 0 r g and r 0 by he definiion of r, and since g(x 0 ) R 1 (Ω, P ) by (4), Lebesgue s convergence heorem can be
4 100 R. S A T O applied o obain lim {M r (x 0 )>a} 1 a M r (x 0 ) dp lim {B r (x 0 )>a} ( 1 a B r (x 0 ) log B r ) (x 0 ) dp = 0 a for all a > 0. Thus, immediaely, lim M r (x 0 )dp = 0. Since < s implies M r (x 0 ) > M r s (x 0 ) 0, i follows ha (8) lim E{M r (x 0 ) x 0 } = 0 a.e. (P ). Furher, since r L 2, i follows from Akcoglu s resul [1] (see also [2]) ha (9) lim n T 1... T nt n... T 1 r exiss a.e. on X. Consequenly, lim sup f n f m n,m lim 2 lim sup e n e m + lim sup g n g m n,m n,m sup e n + lim sup T1... T n nt n... T 1 r T1... TmT m... T 1 r n,m +2 lim sup T1... T nt n... T 1 (g r ) n 2ε + 2 lim sup τ1... τ nτ n... τ 1 (g r ) n (by (2) and (9)) 2ε + 2E{M r (x 0 ) x 0 } (by (5), (6) and (7)); and (8) shows ha (f n (x)), n = 1, 2,..., is a Cauchy sequence for almos all x in X; hus lim n f n (x) exiss almos everywhere, compleing he proof. REFERECES [1] M. A. Akcoglu, Alernaing sequences wih nonposiive operaors, Proc. Amer. Mah. Soc. 104 (1988), [2] M. A. Akcoglu and L. Sucheson, Poinwise convergence of alernaing sequences, Canad. J. Mah. 40 (1988), [3] I. Assani, Alernaing procedures in uniformly smooh Banach spaces, Proc. Amer. Mah. Soc. 104 (1988), [4] D. L. Burkholder, Successive condiional expecaions of an inegrable funcion, Ann. Mah. Sais. 33 (1962), [5] J. L. Doob, A raio operaor limi heorem, Z. Wahrsch. Verw. Gebiee 1 (1963), [6]. A. Fava, Weak ype inequaliies for produc operaors, Sudia Mah. 42 (1972),
5 DUFORD SCHWARTZ OPERATORS 101 [7] U. Krengel, Ergodic Theorems, de Gruyer, Berlin [8] J. e v e u, Discree-Parameer Maringales, orh-holland, Amserdam [9] R. S a o, Ergodic heorems for d-parameer semigroups of Dunford Schwarz operaors, Mah. J. Okayama Univ. 23 (1981), [10]. Sarr, Operaor limi heorems, Trans. Amer. Mah. Soc. 121 (1966), DEPARTMET OF MATHEMATICS, SCHOOL OF SCIECE OKAYAMA UIVERSITY OKAYAMA, 700 JAPA Reçu par la Rédacion le ; en version modifiée le
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