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

## Abstract

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

MSC:39A10, 39B72, 47H10, 46B03.

## 1 Introduction

In 1938, Yosida  proved the following mean ergodic theorem for linear operators: Let E be a real Banach space and $T j$ ($j=1,2,…$) be linear operators of E into itself such that there exists a constant C with $∥( T 1 n ,…, T j n )∥≤C$ for $n=1,2,3,…$ , and $T j$ is weakly completely continuous, i.e., $T j$ maps the closed unite ball of E into a weakly compact subset of E. Then the Cesaro means

$S n , j x= 1 n ∑ k = 1 n T j k x$

converges strongly as $n→+∞$ to a fixed point of $T j$ for each $x∈E$.

On the other hand, in 1975, Baillon  proved the following nonlinear ergodic theorem. Let X be a Banach space and C be a closed convex subset of X. The mappings $T j :C→C$ ($j=1,2,…$) are called nonexpansive on C if

$∥ T j x− T j y∥≤∥x−y∥∀x,y∈C.$

Let $F( T j )$ be the set of fixed points of $T j$. If X is strictly convex, $F( T j )$ is closed and convex. In , Baillon proved the first nonlinear ergodic theorem such that if X is a real Hilbert space and $F( T j )≠∅$, then for each $x∈C$, the sequence ${ S n , j x}$ defined by

$S n , j x= ( 1 n ) ( x + T j x + ⋯ + T j n − 1 x )$

converges weakly to a fixed point of $T j$. It was also shown by Pazy  that if X is a real Hilbert space and $S n , j x$ converges weakly to $y∈C$, then $y∈F( T j )$.

Recently, Rodé  and Takahashi  tried to extend this nonlinear ergodic theorem to semigroup, generalizing the Cesaro means on $N={1,2,…}$, 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 [7, 8]. 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 let $k∈N$. We denote by $E k$ the linear space $E⊕⋯⊕E$ consisting of k-tuples $( x 1 ,…, x k )$, where $x 1 ,…, x k ∈E$. The linear operations on $E k$ are defined coordinate-wise. The zero element of either E or $E k$ is denoted by 0. We denote by $N k$ the set ${1,2,…,k}$ and by $Σ k$ the group of permutations on k symbols.

Definition 2.1 Let E be a linear space, and take $k∈N$. For $σ∈ Σ k$, define

$A σ (x)=( x σ ( 1 ) ,…, x σ ( k ) ),x=( x 1 ,…, x k )∈ E k .$

For $α=( α i )∈ C k$, define

$M α (x)=( α i x i ),x=( x 1 ,…, x k )∈ E k .$

Definition 2.2 Let $(E,∥⋅∥)$ be complex (respectively, real) normed space, and take $n∈N$. A multi-norm of level n on ${ E k :k∈ N n }$ is a sequence $( ∥ ⋅ ∥ k :k∈ N n )$ such that $∥⋅∥$ is a norm on $E k$ for each $k∈ N n$, such that $∥ x ∥ 1 =∥x∥$ for each $x∈E$ (so that $∥ ⋅ ∥ 1$ is the initial norm), and such that the following axioms (A1)-(A4) are satisfied for each $k∈ N n$ with $k≥2$:

• (A1) for each $σ∈ Σ k$ and $x∈ E k$, we have

$∥ A σ ( x ) ∥ k = ∥ x ∥ k ;$
• (A2) for each $α 1 ,…, α k ∈C$ (respectively, each $α 1 ,…, α k ∈R$) and $x∈ E k$, we have

$∥ M α ( x ) ∥ k ≤ ( max i ∈ N k | α i | ) ∥ x ∥ k ;$
• (A3) for each $x 1 ,…, x k − 1$, we have

$∥ ( x 1 , … , x k − 1 , 0 ) ∥ k = ∥ ( x 1 , … , x k − 1 ) ∥ k − 1 ;$
• (A4) for each $x 1 ,…, x k − 1 ∈E$

$∥ ( x 1 , … , x k − 2 , x k − 1 , x k − 1 ) ∥ k = ∥ ( x 1 , … , x k − 1 , x k − 1 ) ∥ k − 1 .$

In this case, $(( E k , ∥ ⋅ ∥ k ):k∈ N n )$ is a multi-normed space of level n.

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

$( ∥ ⋅ ∥ k ) = ( ∥ ⋅ ∥ k : k ∈ N )$

such that $( ∥ ⋅ ∥ k :k∈ N n )$ is a multi-norm of level n for each $n∈N$. In this case, $(( E n , ∥ ⋅ ∥ n ):n∈N)$ is a multi-normed space.

Lemma 2.3 

Suppose that $(( E k , ∥ ⋅ ∥ k ):k∈N)$ is a multi-normed space, and take $k∈ N n$. Then

1. (a)

$∥ ( x , … , x ) ∥ k =∥x∥$ ($x∈E$);

2. (b)

$max i ∈ N k ∥ x i ∥≤ ∥ ( x 1 , … , x k ) ∥ k ≤ ∑ i = 1 k ∥ x i ∥≤k max i ∈ N k ∥ x i ∥$ ($x 1 ,…, x k ∈E$).

It follows from (b) that, if $(E,∥⋅∥)$ is a Banach space, then $( E k , ∥ ⋅ ∥ k )$ is a Banach space for each $k∈N$; in this case $(( E k , ∥ ⋅ ∥ k ):k∈N)$ is a multi-Banach space.

Now we state two important examples of multi-norms for an arbitrary normed space E; cf. .

Example 2.4 The sequence $( ∥ ⋅ ∥ k :k∈N)$ on ${ E k :k∈N}$ defined by

$∥ ( x 1 , … , x k ) ∥ k := max i ∈ N k ∥ x i ∥( x 1 ,…, x k ∈E)$

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

Example 2.5 Let ${( ∥ ⋅ ∥ k α :k∈N):α∈A}$ be the (non-empty) family of all multi-norms on ${ E k :k∈N}$. For $k∈N$, set

$∥ ( x 1 , … , x k ) ∥ k := sup α ∈ A ∥ ( x 1 , … , x k ) ∥ k α ( x 1 ,…, x k ∈E).$

Then $( ∥ ⋅ ∥ k :k∈N)$ is a multi-norm on ${ E k :k∈N}$, called the maximum multi-norm.

We need the following observation, which can easily be deduced from the triangle inequality for the norm $∥ ⋅ ∥ k$ and the property (b) of multi-norms.

Lemma 2.6 Suppose that $k∈N$ and $( x 1 ,…, x k )∈ E k$. For each $j∈{1,…,k}$, let $( x n j ) n = 1 , 2 , …$ be a sequence in E such that $lim n → ∞ x n j = x j$. Then for each $( y 1 ,…, y k )∈ E k$ we have

$lim n → ∞ ( x n 1 − y 1 , … , x n k − y k ) =( x 1 − y 1 ,…, x k − y k ).$

Definition 2.7 Let $(( E k , ∥ ⋅ ∥ k ):k∈N)$ be a multi-normed space. A sequence $( x n )$ in E is a multi-null sequence if, for each $ε>0$, there exists $n 0 ∈N$ such that

$sup k ∈ N ∥ ( x n , … , x n + k − 1 ) ∥ k <ε(n≥ n 0 ).$

Let $x∈E$. We say that the sequence $( x n )$ is multi-convergent to $x∈E$ and write

$lim n → ∞ x n =x$

if $( x n −x)$ is a multi-null sequence.

## 3 Preliminaries and lemmas

Let E a real Banach space and let $E ∗$ be the conjugate space of E, that is, the space of all continuous linear functionals on E. The value of $x ∗ ∈ E ∗$ at $x∈E$ will be denoted by $〈x, x ∗ 〉$. We denote by coD the convex hull of D, $co ¯ D$ the closure of coD.

Let U be a linear continuous operator of E into itself. Then we denote by $U ∗$ the conjugate operator of U.

Assumption (A) Let $( E j , ∥ ⋅ ∥ j )$ be a multi-Banach space and ${ T j , t :t∈G}$ ($j=1,2,…$) be a family of linear continuous operators of a real Banach space $E j$ into itself such that there exists a real number C with $∥ ( T 1 , t , … , T j , t ) ∥ j ≤C$ for all $t∈G$ and the weak closure of ${ T j , t x:t∈G}$ is weakly compact, for each $x∈E$. The index set G is a topological semigroup such that $T j , s t = T j , s ⋅ T j , t$ for all $s,t∈G$ and $T j$ is continuous with respect to the weak operator topology: $〈 T j , s x, x ∗ 〉→〈 T j , t x, x ∗ 〉$ for all $x∈E$ and $x ∗ ∈ E ∗$ if $s→t$ in G.

We denote by $m j (G)$ the Banach space of all bounded continuous real valued functions on the topological semigroup G with the supremum norm. For each $s∈G$ and $f j ∈ m j (G)$, we define elements $l s f j$ and $r s f j$ in $m j (G)$ given by $l s f j (t)= f j (st)$ and $r s f j (t)= f j (ts)$ for all $t∈G$. An element $μ j ∈ m j ( G ) ∗$ (the conjugate space of $m j (G)$) is called a mean on G if $∥ μ j ∥= μ j (1)=1$ moreover, we have $∥ ( μ 1 , … , μ j ) ∥ j = ∑ i = 1 j μ i ( 1 ) j =1$. A mean $μ j$ on G is called left (right) invariant if $μ j ( l s f j )= μ j ( f j )$ ($μ j ( r s f j )= μ j ( f j )$) for all $f j ∈ m j (G)$ and $s∈G$. An invariant mean is a left and right invariant mean. We know that $μ j ∈ m j ( G ) ∗$ is a mean on G if and only if

$inf { f j ( t ) : t ∈ G } ≤ μ j ( f j )≤sup { f j ( t ) : t ∈ G }$

for every $f j ∈ m j (G)$; see [6, 1720].

Let ${ T j , t :t∈G}$ be a family of linear continuous operators of E into itself satisfying Assumption (A) and $μ j$ be a mean on G. Fix $x∈E$. Then, for $x ∗ ∈ E ∗$, the real valued function $t→〈 T j , t x, x ∗ 〉$ is in $m j (G)$. Denote by $μ j , t 〈 T j , t x, x ∗ 〉$ the value of $μ j$ at this function. By linearity of $μ j$ and of $〈⋅,⋅〉$, this is linear in $x ∗$; moreover, since

$| ( μ 1 , t 〈 T 1 , t x , x ∗ 〉 , … , μ j , t 〈 T j , t x , x ∗ 〉 ) | ≤ ∥ ( μ 1 , … , μ j ) ∥ j ⋅ sup t | ( 〈 T 1 , t x , x ∗ 〉 , … , μ j , t 〈 T j , t x , x ∗ 〉 ) | ≤ sup t ∥ ( T 1 x , … , T j x ) ∥ j ⋅ ∥ x ∗ ∥ j ≤ C ⋅ ∥ x ∥ j ⋅ ∥ x ∗ ∥ j$

it is continuous in $x ∗$. Hence we find that $μ j , t 〈 T j , t x,⋅〉$ is an element of $E ∗ ∗$. So, from weak compactness of $co ¯ { T j , t x:t∈G}$ such that $μ j , t 〈 T j , t x, x ∗ 〉=〈 T j , μ j x, x ∗ 〉$ for every $x ∗ ∈ E ∗$.

Put $K= co ¯ { T j , t x:t∈G}$ and suppose that the element $μ j , t 〈 T j , t x,⋅〉$ is not contained in the $n(K)$, where n is the natural embedding of the Banach space E into its second conjugate space $E ∗ ∗$. Then, since the convex set $n(K)$ is compact in the weak topology of $E ∗ ∗$, there exists an element $y ∗ ∈ E ∗$ such that

$μ j , t 〈 T j , t x , y ∗ 〉

Hence, we have

$μ j , t 〈 T j , t x , y ∗ 〉 < inf { 〈 y ∗ , z ∗ ∗ 〉 : z ∗ ∗ ∈ n ( k ) } ≤ inf { 〈 T j , t x , y ∗ 〉 : t ∈ G } ≤ μ j , t 〈 T j , t x , y ∗ 〉 .$

This is a contradiction. Thus, for a mean $μ j$ on G, we can define a linear continuous operator $T j , μ j$ of E into itself such that $∥ ( T 1 , μ 1 , … , T j , μ j ) ∥ j ≤C$, $T j , μ j x∈ co ¯ { T j , t x:t∈G}$ for all $x∈E$, and $μ j , t 〈 T j , t x, x ∗ 〉=〈 T j , μ j x, x ∗ 〉$ for all $x∈E$ and $x ∗ ∈ E ∗$. We denote by $F j (G)$ the set all common fixed points of the mappings $T j , t$, $t∈G$.

Lemma 3.1 Assume that a left invariant mean $μ j$ exists on G, then $T j , μ j (E)⊂ F j (G)$. Especially, $F j (G)$ is then not empty.

Proof Let $x∈E$ and μ be a left invariant mean on G. Then since, for $s∈G$ and $x ∗$,

$〈 T j , s T j , μ J x , x ∗ 〉 = 〈 T j , μ j x , T j , s ∗ x ∗ 〉 = μ j , t 〈 T j , t x , T j , s ∗ x ∗ 〉 = μ j , t 〈 T j , s T j , t x , x ∗ 〉 = μ j , t 〈 T j , s t x , x ∗ 〉 = μ j , t 〈 T j , t x , x ∗ 〉 = 〈 T j , μ j x , x ∗ 〉 ,$

we have $T j , s T j , μ j x= T j , μ j x$. Hence, $T j , μ j (E)⊂ F j (G)$. □

Lemma 3.2 Let $λ j$ be an invariant mean on G. Then $T j , λ j T j , s = T j , s T j , λ j = T j , λ j$ for each $s∈G$ and $T j , λ j T j , μ j = T j , μ j T j , λ j = T j , λ j$ for each mean $μ j$ on G. Especially, $T j , λ j$ is a projection of E onto $F(G)$.

Proof Let $s∈G$. Then, since

$〈 T j , λ j T j , s x , x ∗ 〉 = λ j , t 〈 T j , t T j , s x , x ∗ 〉 = λ j , t 〈 T j , t s x , x ∗ 〉 = λ j , t 〈 T j , t x , x ∗ 〉 = 〈 T j , λ j x , x ∗ 〉$

for $x∈E$ and $x ∗ ∈ E ∗$, we have $T j , λ j T j , s = T j , λ j$. It is obvious from Lemma 3.1 that $T j , s T j , λ j = T j , λ j$ for each $s∈G$. Let $μ j$ be a mean on G. Then, since

$〈 T j , μ j T j , λ j x , x ∗ 〉 = μ j , t 〈 T j , t T j , λ j x , x ∗ 〉 = 〈 μ j , t T j , λ j x , x ∗ 〉 = 〈 T j , λ j x , x ∗ 〉$

and

$〈 T j , λ j T j , μ j x , x ∗ 〉 = 〈 T j , μ j x , T j , λ j ∗ x ∗ 〉 = μ j , t 〈 T j , t x , T j , λ j ∗ x ∗ 〉 = μ j , t 〈 T j , λ j T j , t x , x ∗ 〉 = μ j , t 〈 T j , λ j x , x ∗ 〉 = 〈 T j , λ j x , x ∗ 〉$

for $x∈E$ and $x ∗ ∈ E ∗$, we have $T j , μ j T j , λ j = T j , λ j T j , μ j = T j , λ j$. Putting $μ j = λ j$, we have $T λ j 2 = T λ j$ and hence $T λ j$ is a projection of E onto $F j (G)$. □

As direct consequence of Lemma 3.2, we have the following.

Lemma 3.3 Let $μ j$ and $λ j$ be invariant means on G. Then $T j , μ j = T j , λ j$.

Lemma 3.4 Assume that an invariant mean exists on G. Then, for each $x∈E$, the set $co ¯ { T j , t x:t∈G}∩ F j (G)$ consists of a single point.

Proof Let $x∈E$ and $μ j$ be an invariant mean on G. Then we know that $T j , μ j x∈ F j (G)$ and $T j , μ j x∈ co ¯ { T j , t x:t∈G}$. So, we show that $co ¯ { T j , t x:t∈G}∩ F j (G)={ T j , μ j x}$. Let $x 0 ∈ co ¯ { T j , t x:t∈G}∩ F j (G)$ and $ϵ>0$. Then, for $x ∗ ∈ E ∗$, there exists an element $∑ i = 1 n α i T j , t i x$ in the set $co{ T j , t x:t∈G}$ such that $ϵ>C⋅ ∥ x ∗ ∥ j ⋅ ∥ ∑ i = 1 n α i T j , t i x − x 0 ∥ j$. Hence, we have

$ϵ > C ⋅ ∥ x ∗ ∥ j ⋅ ∥ ∑ i = 1 n α i T j , t i x − x 0 ∥ j ≥ sup t ∥ T j , t ∥ j ⋅ ∥ ∑ i = 1 n α i T j , t i x − x 0 ∥ j ⋅ ∥ x ∗ ∥ j ≥ sup t ∥ ∑ i = 1 n α i T j , t T j , t i x − x 0 ∥ j ⋅ ∥ x ∗ ∥ j ≥ | 〈 ∑ i = 1 n α i T j , t T j , t i x − x 0 , x ∗ 〉 | = | ∑ i = 1 n α i μ j , t 〈 T j , t t i x − x 0 , x ∗ 〉 | = | μ j , t 〈 T j , t x − x 0 , x ∗ 〉 | = | 〈 T j , μ j x − x 0 , x ∗ 〉 | .$

Since ϵ is arbitrary, we have $〈 T j , μ j x, x ∗ 〉=〈 x 0 , x ∗ 〉$ for every $x ∗ ∈ E ∗$ and hence $T j , μ j x= x 0$. □

## 4 Ergodic theorems

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

Theorem 4.1 Let ${ T j , t :t∈G}$ be a family of linear continuous operators in a real Banach space E satisfying Assumption (A). If a net ${ μ j α :α∈I}$ of means on G is asymptotically invariant, i.e.,

$μ j α − r s ∗ μ j α and μ j α − l s ∗ μ j α$

converge to 0 in the weak topology of $m j ( G ) ∗$ for each $s∈G$, then there exists a projection $Q j$ of E on to $F j (G)$ such that $∥ ( Q 1 , … , Q j ) ∥ j ≤C$, $T j , μ j α x$ converges weakly to $Q j x$ for each $x∈E$, $Q j T j , t = T j , t Q j = Q j$ for each $t∈G$, and $Q j x∈ co ¯ { T j , t x:t∈G}$ for each $x∈E$. Furthermore, the projection $Q j$ onto $F j (G)$ is the same for all asymptotically invariant nets.

Proof Let $μ j$ be a cluster point of net ${ μ j α :α∈I}$ in the weak topology of $m j ( G ) ∗$. Then $μ j$ is an invariant mean on G. Hence, by Lemma 3.2, $T j , μ j$ is a projection of E onto $F j (G)$ such that $∥ ( T 1 , μ 1 , … , T j , μ j ) ∥ j ≤C$, $T j , μ j T j , t = T j , t T j , μ j = T j , μ j$ for each $t∈G$ and $T j , μ j x∈ co ¯ { T j , t x:t∈G}$ for each $x∈E$. Setting $Q j = T j , μ j$, we show that $T j , μ j α x$ converges weakly to $Q j x$ for each $x∈E$. Since $T j , μ j α x∈ co ¯ { T j , t x:t∈G}$ for all $α∈I$ and $co ¯ { T j , t x:t∈G}$ is weakly compact, there exists a subnet ${ T j , μ j β x:β∈J}$ of ${ T j , μ j α x:α∈I}$ such that $T j , μ j β x$ converges weakly to an element $x 0 ∈ co ¯ { T j , t x:t∈G}$. To show that $T j , μ j α x$ converges weakly to $Q j x$, it is sufficient to show $x 0 = Q j x$. Let $x ∗ ∈ E ∗$ and $s∈G$. Since $T j , μ j β x→ x 0$ weakly, we have $μ j , t β 〈 T j , t x, x ∗ 〉→〈 x 0 , x ∗ 〉$ and $μ j , t β 〈 T j , t x, T j , s ∗ x ∗ 〉→〈 x 0 , T j , s ∗ x ∗ 〉=〈 T j , s x 0 , x ∗ 〉$. On the other hand, since $μ j β − l s ∗ μ j β →0$ in the weak topology, we have

$μ j , t β 〈 T j , t x , x ∗ 〉 − l s ∗ μ j , t β 〈 T j , t x , x ∗ 〉 = μ j , t β 〈 T j , t x , x ∗ 〉 − μ j , t β 〈 T j , s t x , x ∗ 〉 = μ j , t β 〈 T j , t x , x ∗ 〉 − μ j , t β 〈 T j , t x , T j , s ∗ x ∗ 〉 → 0 .$

Hence, we have $〈 x 0 , x ∗ 〉=〈 T j , s x 0 , x ∗ 〉$ and hence $x 0 ∈ F j (G)$. So, we obtain $Q j x= T j , μ j x= x 0$ from Lemma 3.4. That the projection $Q j$ is the same for all asymptotically invariant nets is obvious from Lemma 3.3. □

As direct consequence of Theorem 4.1, we have the following.

Corollary 4.2 Let ${ T j , t :t∈G}$ be as in Theorem  4.1 and assume that an invariant mean exists on G. Then there exists a projection $Q j$ of E onto $F j$ such that $∥ ( Q 1 , … , Q j ) ∥ j ≤C$, $Q j T j , t = T j , t Q j = Q j$ for each $t∈G$ and $Q j x∈ co ¯ { T j , t x:t∈G}$ for each $x∈E$.

Theorem 4.3 Let ${ T j , t :t∈G}$ be as in Theorem  4.1. If a net ${ μ j α :α∈I}$ of means on G is asymptotically invariant and further $μ j α − r s ∗ μ j α$ converges to 0 in the strong topology of $m j ( G ) ∗$, then there exists a projection $Q j$ of E onto $F j (G)$ such that $∥ ( Q 1 , … , Q j ) ∥ j ≤C$, $T j , μ j α x$ converges strongly to $Q j x$ for each $x∈E$, $Q j T j , t = T j , t Q j = Q j$ for each $t∈G$, and $Q j x∈ co ¯ { T j , t x:t∈G}$ for each $x∈E$.

Proof As in the proof of Theorem 4.1, let $Q j = T j , μ j$, where $μ j$ is a cluster point of the net ${ μ j α :α∈I}$ in the weak topology of $m j ( G ) ∗$. We show that $T j , μ j α x$ converges strongly to $Q j x$ for each $x∈E$.

Let $E 0 = co ¯ {y− T j , t y:y∈E,t∈G}$. Then, for any $z∈ E 0$, $T j , μ j α z$ converges strongly to 0. In fact, if $z=y− T j , s y$, then since, for any $y ∗ ∈ E ∗$,

$| 〈 T j , μ j α z , y ∗ 〉 | = | μ j , t α 〈 T j , t ( y − T j , s y ) , y ∗ 〉 | = | μ j , t α 〈 T j , t y , y ∗ 〉 − μ j , t α 〈 T j , t s y , y ∗ 〉 | = | ( μ j , t α − r s ∗ μ j , t α ) 〈 T j , t y , y ∗ 〉 | ≤ ∥ ( μ 1 α − r s ∗ μ 1 α , … , μ j α − r s ∗ μ j α ) ∥ j ⋅ sup t | 〈 T j , t y , y ∗ 〉 | ≤ ∥ ( μ 1 α − r s ∗ μ 1 α , … , μ j α − r s ∗ μ j α ) ∥ j ⋅ C ⋅ ∥ y ∥ j ⋅ ∥ y ∗ ∥ j ,$

we have $∥ ( T 1 , μ 1 α z , … , T j , μ j α z ) ∥ j ≤C⋅ ∥ ( μ 1 α − r s ∗ μ 1 α , … , μ j α − r s ∗ μ j α ) ∥ j ⋅ ∥ y ∥ j$. Using this inequality, we show that $T j , μ j α z$ converges strongly to 0 for any $z∈ E 0$. Let z be any element of $E 0$ and ϵ be any positive number. By the definition of $E 0$, there exists an element $∑ i = 1 n a i ( y i − T j , s i y i )ϵ$ in the set $co{y− T j , s y:y∈E,s∈G}$ such that $ϵ>2C⋅ ∥ ( z − ∑ i = 1 n a i ( y i − T 1 , s i y i ) , … , z − ∑ i = 1 n a i ( y i − T j , s i y i ) ) ∥ j$. On the other hand, from $∥ ( μ 1 α − r s ∗ μ 1 α , … , μ j α − r s ∗ μ j α ) ∥ j →0$ for all $s∈G$, there exists $a 0 ∈I$ such that, for all $α≥ α 0$ and $i=1,2,…,n$,

$ϵ> ∥ ( μ 1 α − r s i ∗ μ 1 α , … , μ j α − r s i ∗ μ j α ) ∥ j ⋅2C ∥ y i ∥ j .$

This yields

$∥ ( T 1 , μ 1 α z , … , T j , μ j α z ) ∥ j ≤ ∥ ( T 1 , μ 1 α z − T 1 , μ 1 α ( ∑ i = 1 n a i ( y i − T 1 , s i y i ) ) , … , T j , μ j α z − T j , μ j α ( ∑ i = 1 n a i ( y i − T j , s i y i ) ) ) ∥ j + ∥ ( T 1 , μ 1 α ( ∑ i = 1 n a i ( y i − T 1 , s i y i ) ) , … , T j , μ j α ( ∑ i = 1 n a i ( y i − T j , s i y i ) ) ) ∥ j ≤ ∥ ( T 1 , μ 1 α , … , T j , μ j α ) ∥ j ⋅ ∥ ( z − ∑ i = 1 n a i ( y i − T 1 , s i y i ) , … , z − ∑ i = 1 n a i ( y i − T j , s i y i ) ) ∥ j + ∑ i = 1 n ∥ ( T 1 , μ j α ( y i − T 1 , s i y i ) , … , T j , μ j α ( y i − T j , s i y i ) ) ∥ j ≤ C ⋅ ∥ ( z − ∑ i = 1 n a i ( y i − T 1 , s i y i ) , … , z − ∑ i = 1 n a i ( y i − T j , s i y i ) ) ∥ j + sup i ∥ ( μ 1 α − r s i ∗ μ 1 α , … , μ j α − r s i ∗ μ j α ) ∥ j ⋅ C ⋅ ∥ y i ∥ j < ϵ 2 + ϵ 2 = ϵ .$

Hence, $T j , μ j α Z$ converges strongly to 0 for each $z∈ E 0$.

Next, assume that $x− T j , μ j x$ for some $x∈E$ is not contained in the set $E 0$. Then, by the Hahn-Banach theorem, there exists a linear continuous functional $y ∗$ such that $〈x− T j , μ j x, y ∗ 〉=1$ and $〈z, y ∗ 〉=0$ for all $z∈ E 0$. So since $x− T j , t x∈ E 0$ for all $t∈G$, we have

$〈 x − T j , μ j x , y ∗ 〉 = μ j , t 〈 x − T j , t x , y ∗ 〉 =0.$

This is a contradiction. Hence, $x− T j , μ j$ for all $x∈E$ are contained in $E 0$. Therefore we find that $T j , μ j α x− T j , μ j x= T j , μ j α (x− T j , μ j )$ converges strongly to 0 for all $x∈E$. This completes the proof. □

By using Theorem 4.3, we can obtain the following corollary.

Corollary 4.4 Let E be a real Banach space and $T j$ be a linear operator of E into itself such that exists a constant C with $∥ ( T 1 n , … , T j n ) ∥ j ≤C$ for $n=1,2,…$ , and $T j$ is weakly completely continuous, i.e., $T j$ maps the closed unit ball of E into a weakly compact subset of E. Then there exists a projection $Q j$ of E onto the set $F j (T)$ of all fixed point of $T j$ such that $∥ ( Q 1 , … , Q j ) ∥ j ≤C$, the Cesaro means $S j , n = 1 n ∑ k = 1 n T j k x$ converges strongly to $Q j x$ for each $x∈E$, and $T j Q j = Q j T j = Q j$.

Proof Let $x∈E$. Then, since ${ T j n x:n=1,2,…}= T j ({ T n − 1 x:n=1,2,…})⊂ T j (B(0,∥x∥⋅(c+1)))$, where $B(x,r)$ means the closed ball with center x and radius r, the weak closure of ${ T j n x:n=1,2,…}$ is weakly compact. On the other hand, let $G={1,2,3,…}$ with the discrete topology and $μ j n$ be a mean on G such that $μ j n ( f j )= ∑ i = 1 n ( 1 n ) f j (i)$ for each $f j ∈ m j (G)$. Then it is obvious that $∥ ( μ 1 n − r k ∗ μ 1 n , … , μ j n − r k ∗ μ j n ) ∥ j ≤ 2 k n →0$ for all $k∈G$. So, it follows from Theorem 4.3 that Corollary 4.4 is true. □

If $G=[0,∞)$ with the natural topology, then we obtain the corresponding result.

Corollary 4.5 Let E be a real Banach space and ${ T j , t :t∈[0,∞)}$ be a family of linear operators of E into itself satisfying Assumption (A). Then there exists a projection $Q j$ of E onto $F j (G)$ such that $∥ ( Q 1 , … , Q j ) ∥ j ≤C$, $1 T ∫ 0 T T ∫ j , t xdt$ converges strongly to $Q j x$ for each $x∈E$, and $T j , t Q j = Q j T j , t = Q j$ for each $t∈[0,∞)$.

Remark 4.6 $1 T ∫ 0 T T ∫ j , t xdt$ are weak vector valued integrals with respect to means on $G=[0,∞)$. As in Section IV of Rodé , we can also obtain the strong convergence of the sequences

$(1−r) ∑ k = 1 ∞ r k T j k x,r→1−$

and

$λ ∫ 0 ∞ e − λ t T j , t xdt,λ→0+.$

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## 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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### 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.

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