CHARACTERIZATIONS OF VECTOR-VALUED WEAKLY ALMOST PERIODIC FUNCTIONS

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1 IJMMS 2003:5, PII. S Hndaw Publshng Corp. CHARACTERIZATIONS OF VECTOR-VALUED WEAKLY ALMOST PERIODIC FUNCTIONS CHUANYI ZHANG Receved 10 November 2001 We characterze the wea almost perodcty of a vector-valued, bounded, contnuous functon. We show that f the range of the functon s relatvely wealy compact, then the relatve wea compactness of ts rght orbt s equvalent to that of ts left orbt. At the same tme, we gve the functon some other equvalent propertes Mathematcs Subject Classfcaton: 43A60, 43A Introducton. Let S be a semtopologcal semgroup, let be a Banach space, and let S, ) be the space of bounded contnuous functons from S to wth supremum norm. Let f S, ). The rght left) translate of f by s S s the functon R S f L S f ) such that R S ft)= fts) and L S ft)= fst)) for all t S. The functon f s sad to be wealy almost perodc f ts rght orbt R S f ={R S f : s S} s relatvely wealy compact n S, ). Wedenote by S, ) all such functons. In the case that = C, the complex number feld, we wll omt from our notatons and wrte, for example, S) for S,C). Recently, some authors have nvestgated S, ) and exploted ts applcatons n many areas 1, 2, 4, 5, 6, 7, 8, 9]. However, some questons reman unsolved. For example, 2, Theorems and 4.2.6] gve a number of equvalent propertes for a functon f S) to be wealy almost perodc. It s natural to as f the smlar equvalent propertes are true for a functon n S, ). In ths paper, we nvestgate these problems and gve postve answers. It s shown n 4, Proposton 2.8] that the equvalence of relatve wea compactness for R S f and the left orbt L S f ={L S f : s S} holds f S admts an dentty and the range fs)s relatve compact n. We wll gve an example at the end of the paper to show that the assumptons both on S and on fs) are not essental to get the equvalence. We wll show the equvalence under the assumpton that fs)s relatvely wealy compact. At the same tme, we characterze a vector-valued wealy almost perodc functon by gvng t as many equvalent propertes as a scalar-valued wealy almost perodc functon has. We wll not assume that a semtopologcal semgroup S admts an dentty. In fact, f S has an dentty, we can drop the condton fs), beng ether relatve norm compact or relatvely wealy compact see Remar 3.6b)).

2 296 CHUANYI ZHANG 2. Vector-valued means. To show the man results of the paper n Secton 3, we need some facts of vector-valued means. Unless otherwse mentoned, all the results of ths secton come from 8, Sectons 2 and 3]. Let, be two normed lnear spaces and let be the dual space of. Let, ) be the space of bounded lnear operators from to. Wth the norm topology,, ) s a Banach space. We can also furnsh, ) wth the followng two topologes, both of them mae, ) a locally convex topologcal space 3, VI.1.2, VI.1.3]: 1) the strong operator topology τ s, whch s the weaest topology of, ) relatve to whch the mappng µ µz) :, ), s contnuous for each z ; 2) the wea operator topology τ w, whch s the weaest topology of, ) relatve to whch the mappng µ y µz)] :, ) C, s contnuous for each z and y. For, ), we have the followng topology that also maes, ) a locally convex topologcal space 3, page 476]: 3) the wea operator topology τ w, whch s the weaest topology of, ) relatve to whch the mappng µ µz)]y) :, ) C, s contnuous for each z and y. Let S be a nonempty set, let be a Banach space, and let S, ) be the space of bounded functons from S to wth supremum norm. Let be a subspace of S, ) contanng the constant functons. Defnton 2.1. Ameanµ on s a lnear operator from to such that µf) cofs) for all f, denoted by M ) of all means on. We defne the evaluaton mappng ɛ : S M ) as follows: for s S, ɛs)f = fs), f. The followng proposton comes from 9, Propostons 1.5 and 1.6]. Proposton 2.2. Let be a subspace of S, ) contanng the constant functons. Then for both τ s and τ w, M ) s convex and closed n, ), and coɛs)) s dense n M ). Furthermore, f s such that fs) s relatvely wealy compact n for all f, then M ) s τ w -compact. We embed nto ts double dual space canoncally and let ι ) denote ts canoncal mage n ; smlarly, we embed fs) nto for every f and get a subspace ι ) of S,ι )). A functon of mayberegardedas a functon of ι ), and vce versa. Replacng and by ι ) and n Defnton 2.1, respectvely, we get Mι )). A mean of M ) may be regarded as a mean of Mι )), and vce versa. Ths leads to the followng more general defnton of means. Defnton 2.3. Let be a subspace of S, ). A lnear map µ : s called a w mean on provded µf) co w fs), for all f, where w stands for the wea topology σ, ).Denotebyw M ) the set of

3 VECTOR-VALUED W.A.P. FUNCTIONS 297 all w means on. In the case that S, ), we defne w M ) to be w Mι )). Both Defntons 2.1 and 2.3 wll reduce to the defnton of a scalar-valued mean when = C 1, 2.1.2]. Proposton 2.4. Let be a lnear subspace of S, ).Thenforτ w 1) w M ) s convex and compact; 2) let ɛ be the evaluaton map S w M ), then coɛs)) s dense n w M ). Proposton 2.5. of w M S, )). Every member of w M ) can be extended to a member We call the scalar-valued functon space generated space of. = sp { f ) ] x ) : x,f } 2.1) Proposton 2.6. Let be a lnear subspace of S, ) contanng the constant functons, and let be ts generated space. Then, there s an sometrc τ w -σ, ) homeomorphsm µ ϕ µ : w M ) M ) such that ] µf) x ) = ϕ µ f ) x )] f, x ). 2.2) 3. Man results. A nonempty set S that s a semgroup and also a topologcal space s called a semtopologcal semgroup provded that the maps s ts and s st from S to S are contnuous for all t S. LetS be such a set, and let be a subspace of S, ). Wesay s rght resp., left) translaton nvarant f R S ={R S f : s S, f } resp., L S ={L S f : s S, f } ). We say s translaton nvarant f t s both rght and left translaton nvarant. Let be a translaton nvarant subspace of S, ). Forµ M ), defne T µ : S, ) by T µ fs)= µ L s f ) f, s S) 3.1) and U µ : S, ) by U µ fs)= µ R s f ) f, s S). 3.2) We call T µ U µ ) left rght) ntroverson operator determned by µ. Wewllsay that s left rght) ntroverted f T µ U µ ) for all µ M ). Wewll say that s ntroverted f t s both left and rght ntroverted. Smlarly, we defne an ntroverson operator from to S, ) f s a translaton nvarant subspace of S, ) and µ w M ). To show Theorem 3.2, we need the followng proposton that characterzes wea almost perodcty of a functon n S).

4 298 CHUANYI ZHANG Proposton 3.1 2, 4.2.6]. Let S be a semtopologcal semgroup, let f S), and let = S). Then, the followng statements are equvalent: 1) f S); 2) L S f s relatvely wealy compact n S); 3) the mappng ϕ T ϕ f : M ) S) s σ, ) σ S), S) ) contnuous; 4) the mappng ϕ U ϕ f : M ) S) s σ, ) σ S), S) ) contnuous; 5) for all ϕ,ψ M ), ϕt ψ f)= ψu ϕ f). We wll generalze Proposton 3.1 from scalar-valued functon to vectorvalued functon n the next theorem. We wll use some results of the prevous secton to show the theorem. To mae notatons short, we let ι ) = S,ι ) ), ι ) = S,ι ) ), = S, ). 3.3) As n the paragraph before Defnton 2.3, ι ) s the canoncal mage of n ;anf S, ) and ts correspondng functon n ι ) wll be regarded as same functon. Note that both and ι ) have the same generated space S), the space of bounded scalar-valued functons on S. Theorem 3.2. Let S be a semtopologcal semgroup and let be a Banach space. Let = S, ) and let f S, ) be such that fs) s relatvely wealy compact n. Then the followng statements are equvalent: 1) f, that s, R S f s relatvely wealy compact n S, ); 2) L S f s relatvely wealy compact n S, ); 3) the mappng µ T µ f : w M ) ι ) s τ w -σ ι ),ι ) ) contnuous; 4) the mappng µ U µ f : w M ) ι ) s τ w -σ ι ),ι ) ) contnuous; 5) for all µ,ν w M ), µ T ν f ) = ν U µ f ). 3.4) Proof. Snce fs)s relatvely wealy compact n, the functons T µ f and U µ f are n ι ) for all µ w M ). Let ɛ and ɛ be the evaluaton mappngs on and S), respectvely. Let B = cco τ w ɛs), and let B S) be the unt ball of S).By2, ], B S) = cco w ɛ S), where w stands for σ S), S)). It follows from Propostons 2.42) and 2.6 that B and B S) are τ w -σ S), S)) homeomorphc. Defne V : B by Vµ)= T µ f. 3.5)

5 VECTOR-VALUED W.A.P. FUNCTIONS 299 V s contnuous from τ w to the wea pontwse convergence topology P. For, f {µ α } B and µ B are such that µ α µ n τ w,then Tµα fs) ] x ) = µ α LS f )] x ) µ L S f )] x ) = T µ fs) ] x ) x,s S ). 3.6) Snce B s τ w -compact Proposton 2.41)), we have VB)= cco R S f ), 3.7) where closure n ι ) s taen n the wea pontwse convergence topology P. Now, we show that 1) mples 3). By the Kren-Smulan theorem 2, Theorem A.10], ccor S f) s relatvely wealy compact n ι ), whch n vew of 3.7) mples that VB) s σ ι ), ι ) )-closure of ccor S f) n ι ) and that VB) s σ ι ),ι ) )-compact. Therefore, the wea topology σ ι ),ι ) ) and the topology P concde on VB).So,V s τ w -σ ι ),ι ) ) contnuous on B. To show 3), we defne, for Φ ι ), the lnear functonal Φ T : S) C by Φ T ϕ µ ) = Φ Tµ f ), 3.8) where µ and ϕ µ are as n 2.2). It follows from the τ w -σ ι ),ι ) ) contnuty on B of V that Φ T s σ S), S)) contnuous on B S). By Grothendec s completeness theorem 2, Proposton A.8], Φ T s σ S), S)) contnuous on S). Now, we clam that 3) holds. For, f µ α and µ of w M ) are such that µ α µ n τ w,thenϕ µα ϕ µ n σ S), S)) Proposton 2.6), and therefore Φ T µα f ) = Φ T ϕ µα ) Φ T ϕµ ) = Φ Tµ f ). 3.9) Snce Φ s arbtrary n ι ), we have T µα f T µ f n σ ι ),ι ) ). Thus 3) holds. By Proposton 2.41), B s τ w -compact. If 3) holds, then t follows from 3.7) that R S f s relatvely wealy compact n ι ). Thus 1) holds. So, 1) and 3) are equvalent. Smlarly, we show that 2) and 4) are equvalent. Next, we show that 1) and 3) mply 5). Let µ,ν w M ) and ϕ ν,ϕ µ M S)) be as n 2.2). Snce f s n S, ), f )x ) s n S) for all x.byproposton 3.15), ϕ ν { Tϕµ f ) x )]} = ϕ µ { Uϕν f ) x )]}. 3.10)

6 300 CHUANYI ZHANG It follows from 2.2) and3.10) that ν Tµ f )] x ) = ϕ ν { Tϕµ f ) x )]} = ϕ µ { Uϕν f ) x )]} 3.11) = µ U ν f )] x ). Snce x s arbtrary, we have ν T µ f ) = µ U ν f ) µ,ν w M ) ). 3.12) Thus 5) holds. Smlarly, we show that 2) and 4) mply 5). For x and µ w Mι )), defne µ x : ι ) C by µ x h) = µh) ] x ) h ι ) ). 3.13) Then, µ x ι ).LetD ={µ x : x, x =1, µ w Mι ))}.By the separaton theorem, we show that cco w D = B ι ), the unt ball of ι ), where w stands for σ ι ),ι )). To show that 5) mples 4), we need to show that the mappng µ U µ f : B = cco τ w ɛs) ι ) s τ w -σ ι ),ι ) ) contnuous. That s, we need to show that f µ α B and µ B are such that µ α µ n τ w, and f F B ι ), then F U µα f ) F U µ f ). 3.14) Note that cco w D s the unt ball of ι ).Forν x ccod, we have ν x U µα f ] ν x U µ f ], 3.15) because t follows from 3.4) and 3.13) that ν x U µα f ] = ν U µα f )] x ) = µ α Tν f )] x ) µ T ν f )] x ) = ν U µ f )] x ) 3.16) = ν x U µ f ). Let A ={U µ f : µ B}. Then, A s a bounded subset of ι ). Asnthedscusson for Note ) and ) before Theorem 3 of 5], we regard ccod as a set of bounded functon on A, that s, a subset of A). Then, the wea closure

7 VECTOR-VALUED W.A.P. FUNCTIONS 301 cco w D n ι ) s contaned n the wea closure cco w D n A).Sncecco w D s also norm closed n A), forf cco w D cco w D, we have a sequence of {ν n x n} of ccod such that Fh) νn x nh) ) unformly n h A. Snce ) F Uµαf F Uµ f ) ) F Uµαf νn xn Uµα f ) + νn xn Uµα f ) ν n xn Uµ f ) + νn xn Uµ f ) F U µ f ) 3.18), now 3.14) s a consequence of 3.15) and 3.17). Smlarly, we show that 5) mples 3). The proof s complete. Corollary 3.3. Let S,, and be as n Theorem 3.2. Letf. Then, T µ f for all µ M ). Furthermore, f fs) s relatvely wealy compact n, then U µ f for all µ M ). Proof. As n the proof of 1) mplyng 3) of Theorem 3.2, we show the frst statement. If fs) s relatvely wealy compact n, then by the theorem, L S f s relatvely wealy compact n S, ). Note that ths tme f s n. Weshow the second statement as n the proof of 2) mplyng 4). For every x, x =1, and s S, defne x s : S, ) C, by x sf) = x fs) ] f S, ) ). 3.19) Then, x s S, ), the dual space of S, ). SetE ={x s : x, x =1, s S}. LetB = E w, where w stands for the wea topology σ S, ), S, )). Then, B s wea compact. For every f S, ), defne ˆf : E C, by ˆf x s ) = x fs) ] x s E ). 3.20) We extend ˆf from E to B contnuously. So we have ˆf B). Obvously, we have R t f for t S and R t f,thats, R t fx s) = x R t fs)]. The mappng f ˆf : S, ) B) s one to one and lnear sometrc. The space B) s a Banach space wth norm topology. In the next theorem, we wll also equp B) wth the P-topology, the pontwse convergence topology.

8 302 CHUANYI ZHANG Theorem 3.4. Let S be a semtopologcal semgroup, and let f S, ) such that fs)s relatvely wealy compact n. Then, the followng statements are equvalent: 1) f S, ); 2) L S f s relatvely wealy compact n S, ); 3) RS f s relatvely compact n B) n the P-topology; 4) L S f s relatvely compact n B) n the P-topology; 5) lm m lm n x mfs m t n ) = lm n lm m x mfs m t n ), whenever {x m}, x =1, {t n }, {s m } S and all the lmts exst; 6) lm m lm n x mft n s m ) = lm n lm m x mft n s m ), whenever {x m }, x =1, {t n },{s m } S and all the lmts exst. Proof. The equvalence of 1) and 2) comes from Theorem 3.2. If 1) holds, then R S f s relatvely wealy compact n S, ). Sncethemappng f ˆf s one to one and lnear sometrc, R S f s relatvely wealy compact n B). So, R S f s relatvely compact n B) n P-topology. Thus 3) holds. Now, we show 3) mples 5). Let {x m}, x m =1, and {t n },{s m } S be sequences such that the terated lmts lm lm m n x mf ) s m t n, lm lm x n m mf ) s m t n 3.21) exst. Thus, we have { R tn f } B) and {xm s m} E. Letĝ B) be P- topologcal cluster pont of { R tn f }, and let y B be cluster pont of {xm s m }. Then, lm lm R tn f x m n m s m) = lm ĝ x m m s ) m Note that = ĝy) = lm Rtn fy) 3.22) n = lm lm Rtn f x n m m s ) m. lm lm R tn f x ) m n m s m = lm lm x m n mf ) s m t n, lm lm R tn f x ) n m m s m = lm lm x n m mf ) 3.23) s m t n. Therefore, 5) holds. That 5) mples 1) s a consequence of Grothendec s double theorem 2, A.5]. Smlarly, we show the equvalence among 2), 4), and 6). The proof s complete. Example 3.5. Let S = N ={1,2,...}, the semgroup of natural numbers. Let l p,1<p< be the usual sequence spaces wth bass {e n } n=1, where e n ={x } =1 wth x = 1f = n and x = 0 otherwse. Defne f : N l p by fn)= e n n N). 3.24)

9 VECTOR-VALUED W.A.P. FUNCTIONS 303 Snce N s abelan, R S f = L S f. For any subsequences {m }, {n } of S,and{ϕ } of l q wth ϕ 1, where 1/p+1/q = 1 and the norm beng taen n the space l q, we have that lm lm )) lmϕ f m +n = lm )) lmϕ f m +n = lm ) lmϕ em +n = lm ) lmϕ em +n = lm lmϕ,m +n = 0, lmϕ,m +n = 0, 3.25) where ϕ,m +n s the m +n )th component of ϕ. Therefore, by Theorem 3.4 we have f N,l p ). However, fs) s not relatvely norm compact but relatvely wealy compact. Snce R S f = L S f, the two orbts of f are all relatvely wealy compact. We note that N does not admt an dentty. Remar 3.6. a) The equvalence of 1) and 5) of Theorem 2.4 appeared n 5, Theorem 6]; though t was assumed that S admts an dentty, the proof of Theorem 6 dd not use the dentty. b) In both Theorems 3.2 and 3.4, we assume that the range of fs) s relatvely wealy compact to show that the relatve wea compactness of L S f s equvalent to that of R S f.wedonot now f the condton of fs) s essental. We showed n 8, Corollary 8.4] that f S admts an dentty, then fs) s relatvely wealy compact n for all f S, ). Of course, f s reflexve, then any bounded functon has a relatvely wealy compact range. c) From the proof of Theorem 3.4, we see that to get the equvalence among 1), 3), and 5), t does not need the condton fs)beng relatvely wealy compact, nether does the equvalence among 2), 4), and 6). Acnowledgments. The research s supported n part by the Natonal Scence Foundatons NSF) of Chna and Harbn Insttute of Technology HIT). References 1] B. Bast, Some problems concernng dfferent types of vector valued almost perodc functons, Dssertatones Math. Rozprawy Mat.) ), 26. 2] J. F. Berglund, H. D. Junghenn, and P. Mlnes, Analyss on Semgroups. Functon Spaces, Compactfcatons, Representatons, Canadan Mathematcal Socety Seres of Monographs and Advanced Texts, John Wley & Sons, New Yor, ] N. Dunford and J. T. Schwartz, Lnear Operators. I. General Theory, Pure and Appled Mathematcs, vol. 7, Interscence Publshers, New Yor, ] S. Goldberg and P. Irwn, Wealy almost perodc vector-valued functons, Dssertatones Math. Rozprawy Mat.) ), 46. 5] P. Mlnes, On vector-valued wealy almost perodc functons, J. London Math. Soc. 2) ), no. 3, ] W.M.RuessandW.H.Summers,Ergodc theorems for semgroups of operators, Proc. Amer. Math. Soc ), no. 2, ] C. Y. Zhang, Integraton of vector-valued pseudo-almost perodc functons, Proc. Amer. Math. Soc ), no. 1,

10 304 CHUANYI ZHANG 8], Vector-valued means and ther applcatons n some vector-valued functon spaces, Dssertatones Math. Rozprawy Mat.) ), 35. 9], Vector-valued means and wealy almost perodc functons, Int. J. Math. Math. Sc ), no. 2, Chuany Zhang: Department of Mathematcs, Harbn Insttute of Technology, Harbn , Chna E-mal address: czhang@hope.ht.edu.cn

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