Mean ergodic theorem for semigroups of linear operators in multi Banach spaces

R E S E A R C H

Open Access

Mean ergodic theorem for semigroups of

linear operators in multi-Banach spaces

Hassan Morghi Kenari

_{, Reza Saadati}

_{, Mahdi Azhini}

_{and Yeol Je Cho}

*_{Correspondence: [email protected]} 3_{Department of Mathematics}

Education and the RINS, Gyeongsang National University, Jinju, 660-701, Korea

Full list of author information is available at the end of the article

Abstract

In this paper, by using Rodé’s method, we extend Yosida’s theorem to semigroups of linear operators in multi-Banach spaces.

MSC: Primary 39A10; 39B72; secondary 47H10; 46B03

Keywords: ergodic theorem; semigroups; multi-Banach space

1 Introduction

In , Yosida [] proved the following mean ergodic theorem for linear operators: LetE be a real Banach space andTj(j= , , . . .) be linear operators ofEinto itself such that there

exists a constantCwith(T_n, . . . ,T_jn) ≤Cforn= , , , . . . , andTjis weakly completely

continuous,i.e.,Tjmaps the closed unite ball ofEinto a weakly compact subset ofE. Then

the Cesaro means

Sn,jx=

 n

n

k=

T_jkx

converges strongly asn→+∞to a fixed point ofTjfor eachx∈E.

On the other hand, in , Baillon [] proved the following nonlinear ergodic theorem. LetXbe a Banach space andCbe a closed convex subset ofX. The mappingsTj:C→C

(j= , , . . .) are called nonexpansive onCif

Tjx–Tjy ≤ x–y ∀x,y∈C.

LetF(Tj) be the set of fixed points ofTj. IfXis strictly convex,F(Tj) is closed and convex.

In [], Baillon proved the first nonlinear ergodic theorem such that ifXis a real Hilbert space andF(Tj)=∅, then for eachx∈C, the sequence{Sn,jx}defined by

Sn,jx=

 n

x+Tjx+· · ·+Tjn–x

converges weakly to a fixed point ofTj. It was also shown by Pazy [] that ifXis a real

Hilbert space andSn,jxconverges weakly toy∈C, theny∈F(Tj).

Recently, Rodé [] and Takahashi [] tried to extend this nonlinear ergodic theorem to semigroup, generalizing the Cesaro means onN={, , . . .}, such that the corresponding

sequence of mappings converges to a projection onto the set of common fixed points. In this paper, by using Rodé’s method, we extend Yosida’s theorem to semigroups of linear operators in multi-Banach spaces. The proofs employ the methods of Yosida [], Rodé [], Greenleaf [] and Takahashi [, ]. Our paper is motivated from ideas in [].

2 Multi-Banach spaces

The notion of multi-normed space was introduced by Dales and Polyakov in []. This concept is somewhat similar to operator sequence space and has some connections with operator spaces and Banach lattices. Motivations for the study of multi-normed spaces and many examples are given in [–].

Let (E, · ) be a complex normed space, and letk∈N. We denote byEkthe linear space

E⊕ · · · ⊕Econsisting ofk-tuples (x, . . . ,xk), wherex, . . . ,xk∈E. The linear operations

onEkare defined coordinate-wise. The zero element of eitherEorEkis denoted by . We denote byNkthe set{, , . . . ,k}and bykthe group of permutations onksymbols.

Definition . LetEbe a linear space, and takek∈N. Forσ∈k, define

Aσ(x) = (xσ(), . . . ,xσ(k)), x= (x, . . . ,xk)∈Ek.

Forα= (αi)∈Ck, define

Mα(x) = (αixi), x= (x, . . . ,xk)∈Ek.

Definition . Let (E, · ) be complex (respectively, real) normed space, and taken∈N. A multi-norm of levelnon{Ek_:k∈N

n}is a sequence ( · k:k∈Nn) such that · is a

norm onEkfor eachk∈Nn, such thatx=xfor eachx∈E(so that · is the initial

norm), and such that the following axioms (A)-(A) are satisfied for eachk∈Nnwith

k≥:

(A) for eachσ∈kandx∈Ek, we have _Aσ(x)

k=xk;

(A) for eachα, . . . ,αk∈C(respectively, eachα, . . . ,αk∈R) andx∈Ek, we have Mα(x)_k≤

max

i∈Nk

|αi| xk;

(A) for eachx, . . . ,xk–, we have

(x, . . . ,xk–, )_k=(x, . . . ,xk–)_k–;

(A) for eachx, . . . ,xk–∈E

(x, . . . ,xk–,xk–,xk–)_k=(x, . . . ,xk–,xk–)_k–.

In this case, ((Ek_{, ·}

A multi-norm on{Ek_{:k∈N}is a sequence}

· k

= · k:k∈N

such that ( · k:k∈Nn) is a multi-norm of levelnfor eachn∈N. In this case, ((En, · n) :

n∈N) is a multi-normed space.

Lemma .[] Suppose that((Ek_{, ·}

k) :k∈N)is a multi-normed space,and take k∈

Nn.Then

(a) (x, . . . ,x)k=x(x∈E);

(b) maxi∈Nkxi ≤ (x, . . . ,xk)k≤ k

i=xi ≤kmaxi∈Nkxi(x, . . . ,xk∈E).

It follows from (b) that, if (E, · ) is a Banach space, then (Ek_{, ·}

k) is a Banach space

for eachk∈N; in this case ((Ek, · k) :k∈N) is a multi-Banach space.

Now we state two important examples of multi-norms for an arbitrary normed spaceE; cf.[].

Example . The sequence ( · k:k∈N) on{Ek:k∈N}defined by (x, . . . ,xk)_k:=max

i∈Nk

xi (x, . . . ,xk∈E)

is a multi-norm called the minimum multi-norm. The terminology ‘minimum’ is justified by property (b).

Example . Let{( · α

k:k∈N) :α∈A}be the (non-empty) family of all multi-norms on

{Ek_{:k∈N}. Fork∈N, set} (x, . . . ,xk)_k:=sup

α∈A

(x, . . . ,xk) α

k (x, . . . ,xk∈E).

Then ( · k:k∈N) is a multi-norm on{Ek:k∈N}, called the maximum multi-norm.

We need the following observation, which can easily be deduced from the triangle in-equality for the norm · kand the property (b) of multi-norms.

Lemma . Suppose that k∈Nand(x, . . . ,xk)∈Ek.For each j∈ {, . . . ,k},let(xjn)n=,,...

be a sequence in E such thatlimn→∞xjn=xj.Then for each(y, . . . ,yk)∈Ekwe have

lim

n→∞

x_n–y, . . . ,xkn–yk

= (x–y, . . . ,xk–yk).

Definition . Let ((Ek_{, ·}

k) :k∈N) be a multi-normed space. A sequence (xn) inEis

amulti-nullsequence if, for eachε> , there existsn∈Nsuch that

sup

k∈N

(xn, . . . ,xn+k–)_k<ε (n≥n).

Letx∈E. We say that the sequence (xn) ismulti-convergenttox∈Eand write

lim

n→∞xn=x

3 Preliminaries and lemmas

LetEa real Banach space and letE∗be the conjugate space ofE, that is, the space of all continuous linear functionals onE. The value ofx∗∈E∗atx∈Ewill be denoted byx,x∗. We denote bycoDthe convex hull ofD,coDthe closure ofcoD.

LetUbe a linear continuous operator ofEinto itself. Then we denote byU∗the conju-gate operator ofU.

Assumption (A) Let (Ej_{, ·}

j) be a multi-Banach space and{Tj,t:t∈G}(j= , , . . .) be a

family of linear continuous operators of a real Banach spaceEjinto itself such that there

exists a real number Cwith (T,t, . . . ,Tj,t)j≤C for allt∈Gand the weak closure of

{Tj,tx:t∈G}is weakly compact, for eachx∈E. The index setGis a topological semigroup

such thatTj,st=Tj,s·Tj,t for alls,t∈GandTj is continuous with respect to the weak

operator topology:Tj,sx,x∗ → Tj,tx,x∗for allx∈Eandx∗∈E∗ifs→tinG.

We denote bymj(G) the Banach space of all bounded continuous real valued functions

on the topological semigroupGwith the supremum norm. For eachs∈Gandfj∈mj(G),

we define elementslsfjandrsfjinmj(G) given bylsfj(t) =fj(st) andrsfj(t) =fj(ts) for allt∈G.

An elementμj∈mj(G)∗ (the conjugate space ofmj(G)) is called a mean onGifμj=

μj() =  moreover, we have(μ, . . . ,μj)j=

j i=μi()

j = . A meanμjonGis called left

(right) invariant ifμj(lsfj) =μj(fj) (μj(rsfj) =μj(fj)) for allfj∈mj(G) ands∈G. An invariant

mean is a left and right invariant mean. We know thatμj∈mj(G)∗is a mean onGif and

only if

inffj(t) :t∈G

≤μj(fj)≤sup

fj(t) :t∈G

for everyfj∈mj(G); see [, –].

Let {Tj,t :t∈G} be a family of linear continuous operators ofE into itself satisfying

Assumption (A) and μj be a mean on G. Fix x∈E. Then, forx∗∈E∗, the real valued

functiont→ Tj,tx,x∗is inmj(G). Denote byμj,tTj,tx,x∗the value ofμjat this function.

By linearity ofμjand of·,·, this is linear inx∗; moreover, since μ,t

T,tx,x∗

, . . . ,μj,t

Tj,tx,x∗

≤(μ, . . . ,μj)_j·sup t

T,tx,x∗

, . . . ,μj,t

Tj,tx,x∗

≤sup

t

(Tx, . . . ,Tjx)_j·x∗_j

≤C· xj·x∗_j

it is continuous inx∗. Hence we find thatμj,tTj,tx,·is an element ofE∗∗. So, from weak

compactness ofco{Tj,tx:t∈G}such thatμj,tTj,tx,x∗=Tj,μjx,x

∗_{for everyx}∗_∈E∗_.

PutK=co{Tj,tx:t∈G}and suppose that the elementμj,tTj,tx,·is not contained in the

n(K), wherenis the natural embedding of the Banach spaceEinto its second conjugate spaceE∗∗. Then, since the convex setn(K) is compact in theweak∗topology ofE∗∗, there exists an elementy∗∈E∗such that

μj,t

Tj,tx,y∗

Hence, we have

μj,t

Tj,tx,y∗

<infy∗,z∗∗:z∗∗∈n(k)

≤infTj,tx,y∗

:t∈G

≤μj,t

Tj,tx,y∗

.

This is a contradiction. Thus, for a meanμjonG, we can define a linear continuous

op-eratorTj,μj ofEinto itself such that(T,μ, . . . ,Tj,μj)j≤C,Tj,μjx∈co{Tj,tx:t∈G}for all

x∈E, andμj,tTj,tx,x∗=Tj,μjx,x∗for allx∈Eandx∗∈E∗. We denote byFj(G) the set

all common fixed points of the mappingsTj,t,t∈G.

Lemma . Assume that a left invariant meanμjexists on G,then Tj,μj(E)⊂Fj(G).

Espe-cially,Fj(G)is then not empty.

Proof Letx∈Eandμbe a left invariant mean onG. Then since, fors∈Gandx∗,

Tj,sTj,μJx,x

∗_=T

j,μjx,T

∗

j,sx∗

=μj,t

Tj,tx,Tj,s∗x∗

=μj,t

Tj,sTj,tx,x∗

=μj,t

Tj,stx,x∗

=μj,t

Tj,tx,x∗

=Tj,μjx,x

∗_,

we haveTj,sTj,μjx=Tj,μjx. Hence,Tj,μj(E)⊂Fj(G).

Lemma . Letλjbe an invariant mean on G.Then Tj,λjTj,s=Tj,sTj,λj=Tj,λjfor each s∈G

and Tj,λjTj,μj=Tj,μjTj,λj =Tj,λj for each meanμjon G.Especially,Tj,λj is a projection of E

onto F(G).

Proof Lets∈G. Then, since

Tj,λjTj,sx,x

∗_=λ

j,t

Tj,tTj,sx,x∗

=λj,t

Tj,tsx,x∗

=λj,t

Tj,tx,x∗

=Tj,λjx,x

∗

forx∈Eandx∗∈E∗, we haveTj,λjTj,s=Tj,λj. It is obvious from Lemma . thatTj,sTj,λj=

Tj,λj for eachs∈G. Letμjbe a mean onG. Then, since

Tj,μjTj,λjx,x∗

=μj,t

Tj,tTj,λjx,x∗

=μj,tTj,λjx,x∗

=Tj,λjx,x

∗

and

Tj,λjTj,μjx,x

∗_=T

j,μjx,T

∗

j,λjx

∗_=μ

j,t

Tj,tx,Tj,∗λjx

∗

=μj,t

Tj,λjTj,tx,x

∗_=μ

j,t

Tj,λjx,x

∗

=Tj,λjx,x

forx∈Eandx∗∈E∗, we haveTj,μjTj,λj=Tj,λjTj,μj=Tj,λj. Puttingμj=λj, we haveTλj=Tλj

and henceTλj is a projection ofEontoFj(G).

As direct consequence of Lemma ., we have the following.

Lemma . Letμjandλjbe invariant means on G.Then Tj,μj=Tj,λj.

Lemma . Assume that an invariant mean exists on G.Then,for each x∈E, the set co{Tj,tx:t∈G} ∩Fj(G)consists of a single point.

Proof Letx∈E andμj be an invariant mean on G. Then we know that Tj,μjx∈Fj(G)

and Tj,μjx∈co{Tj,tx:t∈G}. So, we show thatco{Tj,tx:t∈G} ∩Fj(G) ={Tj,μjx}. Let

x ∈ co{Tj,tx: t∈ G} ∩Fj(G) and > . Then, for x∗ ∈ E∗, there exists an element n

i=αiTj,tixin the setco{Tj,tx:t∈G}such that>C· x∗j· n

i=αiTj,tix–xj. Hence,

we have

>C·x∗_j·

n

i=

αiTj,tix–x j

≥sup

t

Tj,tj·

n

i=

αiTj,tix–x j

·x∗_j

≥sup

t

n

i=

αiTj,tTj,tix–x j

·x∗_j

≥

_n

i=

αiTj,tTj,tix–x,x∗

=

n

i=

αiμj,t

Tj,ttix–x,x∗

=μj,t

Tj,tx–x,x∗

=Tj,μjx–x,x

∗_.

Since is arbitrary, we haveTj,μjx,x∗=x,x∗for everyx∗∈E∗and henceTj,μjx=x.

4 Ergodic theorems

Now, we can prove mean ergodic theorems for semigroups of linear continuous operators in multi-Banach space.

Theorem . Let{Tj,t:t∈G}be a family of linear continuous operators in a real Banach

space E satisfying Assumption(A).If a net{μα

j :α∈I}of means on G is asymptotically

invariant,i.e.,

μα_j –r_s∗μα_j and μα_j –l∗_sμα_j

converge toin the weak∗topology of mj(G)∗for each s∈G,then there exists a projection Qj

of E on to Fj(G)such that(Q, . . . ,Qj)j≤C,Tj,μα

QjTj,t=Tj,tQj=Qjfor each t∈G,and Qjx∈co{Tj,tx:t∈G}for each x∈E.Furthermore,

the projection Qjonto Fj(G)is the same for all asymptotically invariant nets.

Proof Let μj be a cluster point of net{μαj :α ∈I} in the weak∗ topology of mj(G)∗.

Then μj is an invariant mean on G. Hence, by Lemma ., Tj,μj is a projection of E

ontoFj(G) such that(T,μ, . . . ,Tj,μj)j≤C,Tj,μjTj,t=Tj,tTj,μj=Tj,μj for eacht∈Gand

Tj,μjx∈co{Tj,tx:t∈G}for eachx∈E. SettingQj=Tj,μj, we show thatTj,μαjxconverges

weakly toQjxfor eachx∈E. SinceTj,μα_jx∈co{Tj,tx:t∈G}for allα∈Iandco{Tj,tx:t∈G}

is weakly compact, there exists a subnet{T_j,μβ

jx:

β∈J}of{Tj,μα_jx:α∈I}such thatT_j,μβ

jx

converges weakly to an elementx∈co{Tj,tx:t∈G}. To show thatTj,μα

jxconverges weakly

toQjx, it is sufficient to showx=Qjx. Letx∗∈E∗ands∈G. SinceT_j,μβ

jx→xweakly, we

haveμβ_j,tTj,tx,x∗ → x,x∗andμβj,tTj,tx,Tj,s∗x∗ → x,Tj,s∗x∗=Tj,sx,x∗. On the other

hand, sinceμβ_j –l_s∗μβ_j → in theweak∗topology, we have

μβ_j,tTj,tx,x∗

–l∗_sμβ_j,tTj,tx,x∗

=μβ_j,tTj,tx,x∗

–μβ_j,tTj,stx,x∗

=μβ_j,tTj,tx,x∗

–μβ_j,tTj,tx,Tj,s∗x∗

→.

Hence, we havex,x∗=Tj,sx,x∗and hencex∈Fj(G). So, we obtainQjx=Tj,μjx=x

from Lemma .. That the projectionQjis the same for all asymptotically invariant nets

is obvious from Lemma ..

As direct consequence of Theorem ., we have the following.

Corollary . Let{Tj,t:t∈G}be as in Theorem.and assume that an invariant mean

exists on G.Then there exists a projection Qj of E onto Fj such that(Q, . . . ,Qj)j≤C,

QjTj,t=Tj,tQj=Qjfor each t∈G and Qjx∈co{Tj,tx:t∈G}for each x∈E.

Theorem . Let{Tj,t:t∈G}be as in Theorem..If a net{μαj :α∈I}of means on G

is asymptotically invariant and further μα_j –r_s∗μα_j converges toin the strong topology of mj(G)∗,then there exists a projection Qjof E onto Fj(G)such that(Q, . . . ,Qj)j≤C,

Tj,μα_jx converges strongly to Qjx for each x∈E,QjTj,t=Tj,tQj=Qjfor each t∈G,and Qjx∈

co{Tj,tx:t∈G}for each x∈E.

Proof As in the proof of Theorem ., letQj=Tj,μj, whereμjis a cluster point of the net

{μα

j :α∈I}in theweak∗topology ofmj(G)∗. We show thatTj,μα_jxconverges strongly to

Qjxfor eachx∈E.

LetE=co{y–Tj,ty:y∈E,t∈G}. Then, for anyz∈E,Tj,μα_jzconverges strongly to .

In fact, ifz=y–Tj,sy, then since, for anyy∗∈E∗,

Tj,μα_jz,y∗=μαj,t

Tj,t(y–Tj,sy),y∗

=μα_j,tTj,ty,y∗

–μα_j,tTj,tsy,y∗

≤μα_ –r_s∗μα_, . . . ,μα_j –r∗_sμα_j

j·sup t

Tj,ty,y∗

≤μα_ –r_s∗μα_, . . . ,μα_j –r_s∗μα_jj·C· yj·y∗_j,

we have(T,μα_z, . . . ,Tj,μα_jz)j≤C· (μα–r∗sμ α

, . . . ,μαj –r∗sμ α

j)j· yj. Using this

inequal-ity, we show thatTj,μα_jzconverges strongly to  for anyz∈E. Letzbe any element ofE

andbe any positive number. By the definition ofE, there exists an element n

i=ai(yi–

Tj,siyi)in the setco{y–Tj,sy:y∈E,s∈G}such that> C·(z– n

i=ai(yi–T,siyi), . . . ,z– n

i=ai(yi–Tj,siyi))j. On the other hand, from(μα –r∗sμ α , . . . ,μ

α j –rs∗μ

α

j)j→ for all

s∈G, there existsa∈Isuch that, for allα≥αandi= , , . . . ,n,

>μα_ –r_si∗μα_, . . . ,μα_j –r_si∗μα_jj·Cyij.

This yields

(T,μα_z, . . . ,Tj,μα_jz)_j

≤

T,μα

z–T,μα

_n

i=

ai(yi–T,siyi)

,

. . . ,Tj,μα_jz–Tj,μα_{j n}

i=

ai(yi–Tj,siyi) j +

T,μα_{ n}

i=

ai(yi–T,siyi)

, . . . ,Tj,μα_{j n}

i=

ai(yi–Tj,siyi)

j

≤(T,μα_, . . . ,Tj,μα_j)_j

· z– n i=

ai(yi–T,siyi), . . . ,z– n

i=

ai(yi–Tj,siyi) j + n i=

T,μα_j(yi–T,siyi), . . . ,Tj,μα_j(yi–Tj,siyi)_j

≤C·

z– n i=

ai(yi–T,siyi), . . . ,z– n

i=

ai(yi–Tj,siyi) j +sup i

μα_ –r_si∗μα_, . . . ,μα_j –r∗_siμα_j

j·C· yij

< +

=.

Hence,Tj,μα_jZconverges strongly to  for eachz∈E.

Next, assume thatx–Tj,μjxfor somex∈E is not contained in the setE. Then, by

the Hahn-Banach theorem, there exists a linear continuous functionaly∗ such thatx– Tj,μjx,y∗=  andz,y∗=  for allz∈E. So sincex–Tj,tx∈Efor allt∈G, we have

x–Tj,μjx,y

∗_=μ

j,t

x–Tj,tx,y∗

= .

This is a contradiction. Hence,x–Tj,μjfor allx∈Eare contained inE. Therefore we find

thatTj,μα_jx–Tj,μjx=Tj,μα_j(x–Tj,μj) converges strongly to  for allx∈E. This completes

By using Theorem ., we can obtain the following corollary.

Corollary . Let E be a real Banach space and Tjbe a linear operator of E into itself

such that exists a constant C with (Tn

, . . . ,Tjn)j≤C for n= , , . . . ,and Tj is weakly

completely continuous,i.e.,Tjmaps the closed unit ball of E into a weakly compact subset

of E.Then there exists a projection Qjof E onto the set Fj(T)of all fixed point of Tjsuch that

(Q, . . . ,Qj)j≤C,the Cesaro means Sj,n=_n n

k=Tjkx converges strongly to Qjx for each

x∈E,and TjQj=QjTj=Qj.

Proof Letx∈E. Then, since{T_jnx:n= , , . . .}=Tj({Tn–x:n= , , . . .})⊂Tj(B(,x ·

(c+ ))), whereB(x,r) means the closed ball with centerxand radiusr, the weak closure of{Tn

jx:n= , , . . .}is weakly compact. On the other hand, letG={, , , . . .}with the

discrete topology andμn_j be a mean onGsuch thatμn_j(fj) = n

i=( 

n)fj(i) for eachfj∈mj(G).

Then it is obvious that(μn_–r_k∗μn_, . . . ,μn_j –r∗_kμn_j)j≤k_n → for allk∈G. So, it follows

from Theorem . that Corollary . is true.

IfG= [,∞) with the natural topology, then we obtain the corresponding result.

Corollary . Let E be a real Banach space and{Tj,t :t∈[,∞)} be a family of linear

operators of E into itself satisfying Assumption(A).Then there exists a projection Qjof E

onto Fj(G)such that(Q, . . . ,Qj)j≤C, _T T

 T

j,tx dt converges strongly to Qjx for each

x∈E,and Tj,tQj=QjTj,t=Qjfor each t∈[,∞).

Remark . _T_TT_j,tx dt are weak vector valued integrals with respect to means on G= [,∞). As in Section IV of Rodé [], we can also obtain the strong convergence of the sequences

( –r) ∞

k=

rkT_jkx, r→–

and

λ

_∞



e–λtTj,tx dt, λ→ + .

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors carried out the proof. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics, Science and Research Branch, Islamic Azad University, Ashrafi Esfahani Ave., Tehran, 14778,}

Iran.2_{Department of Mathematics and Computer Science, Iran University of Science and Technology, Tehran, Iran.} 3_{Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju, 660-701, Korea.}

Acknowledgements

Yeol Je Cho was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2013053358).

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Cite this article as:Kenari et al.:Mean ergodic theorem for semigroups of linear operators in multi-Banach spaces.