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 [1] proved the following mean ergodic theorem for linear operators: Let E be a real Banach space and $T_j$ ($j=1,2,\dots$) be linear operators of E into itself such that there exists a constant C with $\parallel (T_1^n,\dots ,T_j^n)\parallel \le C$ for $n=1,2,3,\dots$ , 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

$$S_{n,j}x=\frac{1}{n}\sum _{k=1}^n{T}_j^{k}x$$

converges strongly as $n\to +\mathrm{\infty }$ to a fixed point of $T_j$ for each $x\in E$.

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

$$\parallel T_{j}x-T_{j}y\parallel \le \parallel x-y\parallel \mathrm{\forall }x,y\in C.$$

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 [2], Baillon proved the first nonlinear ergodic theorem such that if X is a real Hilbert space and $F(T_j)\ne \mathrm{\varnothing }$, then for each $x\in C$, the sequence $\{S_{n,j}x\}$ defined by

$$S_{n,j}x=\left(\frac{1}{n}\right)(x+T_{j}x+\cdots +T_j^{n-1}x)$$

converges weakly to a fixed point of $T_j$. It was also shown by Pazy [3] that if X is a real Hilbert space and $S_{n,j}x$ converges weakly to $y\in C$, then $y\in F(T_j)$.

Recently, Rodé [4] and Takahashi [5] tried to extend this nonlinear ergodic theorem to semigroup, generalizing the Cesaro means on $\mathbf{N}=\{1,2,\dots \}$, 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 [1], Rodé [4], Greenleaf [6] and Takahashi [7, 8]. Our paper is motivated from ideas in [9].

2 Multi-Banach spaces

The notion of multi-normed space was introduced by Dales and Polyakov in [10]. 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 [10–16].

Let $(E,\parallel \cdot \parallel )$ be a complex normed space, and let $k\in \mathbf{N}$. We denote by $E^k$ the linear space $E\oplus \cdots \oplus E$ consisting of k-tuples $(x_1,\dots ,x_k)$, where $x_1,\dots ,x_k\in E$. 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 ${\mathbf{N}}_k$ the set $\{1,2,\dots ,k\}$ and by ${\mathrm{\Sigma }}_k$ the group of permutations on k symbols.

Definition 2.1 Let E be a linear space, and take $k\in \mathbf{N}$. For $\sigma \in {\mathrm{\Sigma }}_k$, define

$$A_{\sigma }(x)=(x_{\sigma (1)},\dots ,x_{\sigma (k)}),x=(x_1,\dots ,x_k)\in E^k.$$

For $\alpha =({\alpha }_i)\in {\mathbf{C}}^k$, define

$$M_{\alpha }(x)=({\alpha }_i{x}_i),x=(x_1,\dots ,x_k)\in E^k.$$

Definition 2.2 Let $(E,\parallel \cdot \parallel )$ be complex (respectively, real) normed space, and take $n\in \mathbf{N}$. A multi-norm of level n on $\{E^k:k\in {\mathbf{N}}_n\}$ is a sequence $({\parallel \cdot \parallel }_k:k\in {\mathbf{N}}_n)$ such that $\parallel \cdot \parallel$ is a norm on $E^k$ for each $k\in {\mathbf{N}}_n$, such that ${\parallel x\parallel }_1=\parallel x\parallel$ for each $x\in E$ (so that ${\parallel \cdot \parallel }_1$ is the initial norm), and such that the following axioms (A1)-(A4) are satisfied for each $k\in {\mathbf{N}}_n$ with $k\ge 2$:

• (A1) for each $\sigma \in {\mathrm{\Sigma }}_k$ and $x\in E^k$, we have

  $${\parallel A_{\sigma }(x)\parallel }_k={\parallel x\parallel }_k;$$

• (A2) for each ${\alpha }_1,\dots ,{\alpha }_k\in \mathbf{C}$ (respectively, each ${\alpha }_1,\dots ,{\alpha }_k\in \mathbf{R}$) and $x\in E^k$, we have

  $${\parallel M_{\alpha }(x)\parallel }_k\le (\underset{i\in {\mathbf{N}}_k}{max}|{\alpha }_i|){\parallel x\parallel }_k;$$

• (A3) for each $x_1,\dots ,x_{k-1}$, we have

  $${\parallel (x_1,\dots ,x_{k-1},0)\parallel }_k={\parallel (x_1,\dots ,x_{k-1})\parallel }_{k-1};$$

• (A4) for each $x_1,\dots ,x_{k-1}\in E$

  $${\parallel (x_1,\dots ,x_{k-2},x_{k-1},x_{k-1})\parallel }_k={\parallel (x_1,\dots ,x_{k-1},x_{k-1})\parallel }_{k-1}.$$

In this case, $((E^k,{\parallel \cdot \parallel }_k):k\in {\mathbf{N}}_n)$ is a multi-normed space of level n.

A multi-norm on $\{E^k:k\in \mathbf{N}\}$ is a sequence

$$\left({\parallel \cdot \parallel }_k\right)=({\parallel \cdot \parallel }_k:k\in \mathbf{N})$$

such that $({\parallel \cdot \parallel }_k:k\in {\mathbf{N}}_n)$ is a multi-norm of level n for each $n\in \mathbf{N}$. In this case, $((E^n,{\parallel \cdot \parallel }_n):n\in \mathbf{N})$ is a multi-normed space.

Lemma 2.3 [12]

Suppose that $((E^k,{\parallel \cdot \parallel }_k):k\in \mathbf{N})$ is a multi-normed space, and take $k\in {\mathbf{N}}_n$. Then

• (a) ${\parallel (x,\dots ,x)\parallel }_k=\parallel x\parallel$ ($x\in E$);

• (b) ${max}_{i\in {\mathbf{N}}_k}\parallel x_i\parallel \le {\parallel (x_1,\dots ,x_k)\parallel }_k\le {\sum }_{i=1}^k\parallel x_i\parallel \le k{max}_{i\in {\mathbf{N}}_k}\parallel x_i\parallel$ ($x_1,\dots ,x_k\in E$).

It follows from (b) that, if $(E,\parallel \cdot \parallel )$ is a Banach space, then $(E^k,{\parallel \cdot \parallel }_k)$ is a Banach space for each $k\in \mathbf{N}$; in this case $((E^k,{\parallel \cdot \parallel }_k):k\in \mathbf{N})$ is a multi-Banach space.

Now we state two important examples of multi-norms for an arbitrary normed space E; cf. [10].

Example 2.4 The sequence $({\parallel \cdot \parallel }_k:k\in \mathbf{N})$ on $\{E^k:k\in \mathbf{N}\}$ defined by

$${\parallel (x_1,\dots ,x_k)\parallel }_k:=\underset{i\in {\mathbf{N}}_k}{max}\parallel x_i\parallel (x_1,\dots ,x_k\in E)$$

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

Example 2.5 Let $\{({\parallel \cdot \parallel }_k^{\alpha }:k\in \mathbf{N}):\alpha \in A\}$ be the (non-empty) family of all multi-norms on $\{E^k:k\in \mathbf{N}\}$. For $k\in \mathbf{N}$, set

$${\parallel (x_1,\dots ,x_k)\parallel }_k:=\underset{\alpha \in A}{sup}{\parallel (x_1,\dots ,x_k)\parallel }_k^{\alpha }(x_1,\dots ,x_k\in E).$$

Then $({\parallel \cdot \parallel }_k:k\in \mathbf{N})$ is a multi-norm on $\{E^k:k\in \mathbf{N}\}$, called the maximum multi-norm.

We need the following observation, which can easily be deduced from the triangle inequality for the norm ${\parallel \cdot \parallel }_k$ and the property (b) of multi-norms.

Lemma 2.6 Suppose that $k\in \mathbf{N}$ and $(x_1,\dots ,x_k)\in E^k$. For each $j\in \{1,\dots ,k\}$, let ${(x_n^j)}_{n=1,2,\dots }$ be a sequence in E such that ${lim}_{n\to \mathrm{\infty }}{x}_n^j=x_j$. Then for each $(y_1,\dots ,y_k)\in E^k$ we have

$$\underset{n\to \mathrm{\infty }}{lim}(x_n^1-y_1,\dots ,x_n^k-y_k)=(x_1-y_1,\dots ,x_k-y_k).$$

Definition 2.7 Let $((E^k,{\parallel \cdot \parallel }_k):k\in \mathbf{N})$ be a multi-normed space. A sequence $(x_n)$ in E is a multi-null sequence if, for each $\epsilon >0$, there exists $n_0\in \mathbf{N}$ such that

$$\underset{k\in \mathbf{N}}{sup}{\parallel (x_n,\dots ,x_{n+k-1})\parallel }_k<\epsilon (n\ge n_0).$$

Let $x\in \mathcal{E}$. We say that the sequence $(x_n)$ is multi-convergent to $x\in E$ and write

$$\underset{n\to \mathrm{\infty }}{lim}{x}_n=x$$

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

3 Preliminaries and lemmas

Let E a real Banach space and let $E^{\ast }$ be the conjugate space of E, that is, the space of all continuous linear functionals on E. The value of $x^{\ast }\in E^{\ast }$ at $x\in E$ will be denoted by $〈x,x^{\ast }〉$. We denote by coD the convex hull of D, $\overline{co}D$ the closure of coD.

Let U be a linear continuous operator of E into itself. Then we denote by $U^{\ast }$ the conjugate operator of U.

Assumption (A) Let $(E^j,{\parallel \cdot \parallel }_j)$ be a multi-Banach space and $\{T_{j,t}:t\in G\}$ ($j=1,2,\dots$) 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 ${\parallel (T_{1,t},\dots ,T_{j,t})\parallel }_j\le C$ for all $t\in G$ and the weak closure of $\{T_{j,t}x:t\in G\}$ is weakly compact, for each $x\in E$. The index set G is a topological semigroup such that $T_{j,st}=T_{j,s}\cdot T_{j,t}$ for all $s,t\in G$ and $T_j$ is continuous with respect to the weak operator topology: $〈T_{j,s}x,x^{\ast }〉\to 〈T_{j,t}x,x^{\ast }〉$ for all $x\in E$ and $x^{\ast }\in E^{\ast }$ if $s\to t$ 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 $s\in G$ and $f_j\in m_j(G)$, 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 $t\in G$. An element ${\mu }_j\in m_j{(G)}^{\ast }$ (the conjugate space of $m_j(G)$) is called a mean on G if $\parallel {\mu }_j\parallel ={\mu }_j(1)=1$ moreover, we have ${\parallel ({\mu }_1,\dots ,{\mu }_j)\parallel }_j=\frac{{\sum }_{i=1}^j{\mu }_i(1)}{j}=1$. A mean ${\mu }_j$ on G is called left (right) invariant if ${\mu }_j(l_s{f}_j)={\mu }_j(f_j)$ (${\mu }_j(r_s{f}_j)={\mu }_j(f_j)$) for all $f_j\in m_j(G)$ and $s\in G$. An invariant mean is a left and right invariant mean. We know that ${\mu }_j\in m_j{(G)}^{\ast }$ is a mean on G if and only if

$$inf\{f_j(t):t\in G\}\le {\mu }_j(f_j)\le sup\{f_j(t):t\in G\}$$

for every $f_j\in m_j(G)$; see [6, 17–20].

Let $\{T_{j,t}:t\in G\}$ be a family of linear continuous operators of E into itself satisfying Assumption (A) and ${\mu }_j$ be a mean on G. Fix $x\in E$. Then, for $x^{\ast }\in E^{\ast }$, the real valued function $t\to 〈T_{j,t}x,x^{\ast }〉$ is in $m_j(G)$. Denote by ${\mu }_{j,t}〈T_{j,t}x,x^{\ast }〉$ the value of ${\mu }_j$ at this function. By linearity of ${\mu }_j$ and of $〈\cdot ,\cdot 〉$, this is linear in $x^{\ast }$; moreover, since

$$\begin{array}{c}\left|({\mu }_{1,t}〈T_{1,t}x,x^{\ast }〉,\dots ,{\mu }_{j,t}〈T_{j,t}x,x^{\ast }〉)\right|\hfill \\ \le {\parallel ({\mu }_1,\dots ,{\mu }_j)\parallel }_j\cdot \underset{t}{sup}\left|(〈T_{1,t}x,x^{\ast }〉,\dots ,{\mu }_{j,t}〈T_{j,t}x,x^{\ast }〉)\right|\hfill \\ \le \underset{t}{sup}{\parallel (T_{1}x,\dots ,T_{j}x)\parallel }_j\cdot {\parallel x^{\ast }\parallel }_j\hfill \\ \le C\cdot {\parallel x\parallel }_j\cdot {\parallel x^{\ast }\parallel }_j\hfill \end{array}$$

it is continuous in $x^{\ast }$. Hence we find that ${\mu }_{j,t}〈T_{j,t}x,\cdot 〉$ is an element of $E^{\ast \ast }$. So, from weak compactness of $\overline{co}\{T_{j,t}x:t\in G\}$ such that ${\mu }_{j,t}〈T_{j,t}x,x^{\ast }〉=〈T_{j,{\mu }_j}x,x^{\ast }〉$ for every $x^{\ast }\in E^{\ast }$.

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

$${\mu }_{j,t}〈T_{j,t}x,y^{\ast }〉<inf\{〈y^{\ast },z^{\ast \ast }〉:z^{\ast \ast }\in n(k)\}.$$

Hence, we have

$$\begin{array}{rcl}{\mu }_{j,t}〈T_{j,t}x,y^{\ast }〉& <& inf\{〈y^{\ast },z^{\ast \ast }〉:z^{\ast \ast }\in n(k)\}\\ \le & inf\{〈T_{j,t}x,y^{\ast }〉:t\in G\}\\ \le & {\mu }_{j,t}〈T_{j,t}x,y^{\ast }〉.\end{array}$$

This is a contradiction. Thus, for a mean ${\mu }_j$ on G, we can define a linear continuous operator $T_{j,{\mu }_j}$ of E into itself such that ${\parallel (T_{1,{\mu }_1},\dots ,T_{j,{\mu }_j})\parallel }_j\le C$, $T_{j,{\mu }_j}x\in \overline{co}\{T_{j,t}x:t\in G\}$ for all $x\in E$, and ${\mu }_{j,t}〈T_{j,t}x,x^{\ast }〉=〈T_{j,{\mu }_j}x,x^{\ast }〉$ for all $x\in E$ and $x^{\ast }\in E^{\ast }$. We denote by $F_j(G)$ the set all common fixed points of the mappings $T_{j,t}$, $t\in G$.

Lemma 3.1 Assume that a left invariant mean ${\mu }_j$ exists on G, then $T_{j,{\mu }_j}(E)\subset F_j(G)$. Especially, $F_j(G)$ is then not empty.

Proof Let $x\in E$ and μ be a left invariant mean on G. Then since, for $s\in G$ and $x^{\ast }$,

$$\begin{array}{rcl}〈T_{j,s}{T}_{j,{\mu }_J}x,x^{\ast }〉& =& 〈T_{j,{\mu }_j}x,T_{j,s}^{\ast }{x}^{\ast }〉\\ =& {\mu }_{j,t}〈T_{j,t}x,T_{j,s}^{\ast }{x}^{\ast }〉={\mu }_{j,t}〈T_{j,s}{T}_{j,t}x,x^{\ast }〉\\ =& {\mu }_{j,t}〈T_{j,st}x,x^{\ast }〉={\mu }_{j,t}〈T_{j,t}x,x^{\ast }〉\\ =& 〈T_{j,{\mu }_j}x,x^{\ast }〉,\end{array}$$

we have $T_{j,s}{T}_{j,{\mu }_j}x=T_{j,{\mu }_j}x$. Hence, $T_{j,{\mu }_j}(E)\subset F_j(G)$. □

Lemma 3.2 Let ${\lambda }_j$ be an invariant mean on G. Then $T_{j,{\lambda }_j}{T}_{j,s}=T_{j,s}{T}_{j,{\lambda }_j}=T_{j,{\lambda }_j}$ for each $s\in G$ and $T_{j,{\lambda }_j}{T}_{j,{\mu }_j}=T_{j,{\mu }_j}{T}_{j,{\lambda }_j}=T_{j,{\lambda }_j}$ for each mean ${\mu }_j$ on G. Especially, $T_{j,{\lambda }_j}$ is a projection of E onto $F(G)$.

Proof Let $s\in G$. Then, since

$$\begin{array}{rcl}〈T_{j,{\lambda }_j}{T}_{j,s}x,x^{\ast }〉& =& {\lambda }_{j,t}〈T_{j,t}{T}_{j,s}x,x^{\ast }〉={\lambda }_{j,t}〈T_{j,ts}x,x^{\ast }〉\\ =& {\lambda }_{j,t}〈T_{j,t}x,x^{\ast }〉=〈T_{j,{\lambda }_j}x,x^{\ast }〉\end{array}$$

for $x\in E$ and $x^{\ast }\in E^{\ast }$, we have $T_{j,{\lambda }_j}{T}_{j,s}=T_{j,{\lambda }_j}$. It is obvious from Lemma 3.1 that $T_{j,s}{T}_{j,{\lambda }_j}=T_{j,{\lambda }_j}$ for each $s\in G$. Let ${\mu }_j$ be a mean on G. Then, since

$$\begin{array}{rcl}〈T_{j,{\mu }_j}{T}_{j,{\lambda }_j}x,x^{\ast }〉& =& {\mu }_{j,t}〈T_{j,t}{T}_{j,{\lambda }_j}x,x^{\ast }〉=〈{\mu }_{j,t}{T}_{j,{\lambda }_j}x,x^{\ast }〉\\ =& 〈T_{j,{\lambda }_j}x,x^{\ast }〉\end{array}$$

and

$$\begin{array}{rcl}〈T_{j,{\lambda }_j}{T}_{j,{\mu }_j}x,x^{\ast }〉& =& 〈T_{j,{\mu }_j}x,T_{j,{\lambda }_j}^{\ast }{x}^{\ast }〉={\mu }_{j,t}〈T_{j,t}x,T_{j,{\lambda }_j}^{\ast }{x}^{\ast }〉\\ =& {\mu }_{j,t}〈T_{j,{\lambda }_j}{T}_{j,t}x,x^{\ast }〉={\mu }_{j,t}〈T_{j,{\lambda }_j}x,x^{\ast }〉\\ =& 〈T_{j,{\lambda }_j}x,x^{\ast }〉\end{array}$$

for $x\in E$ and $x^{\ast }\in E^{\ast }$, we have $T_{j,{\mu }_j}{T}_{j,{\lambda }_j}=T_{j,{\lambda }_j}{T}_{j,{\mu }_j}=T_{j,{\lambda }_j}$. Putting ${\mu }_j={\lambda }_j$, we have $T_{{\lambda }_j}^2=T_{{\lambda }_j}$ and hence $T_{{\lambda }_j}$ is a projection of E onto $F_j(G)$. □

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

Lemma 3.3 Let ${\mu }_j$ and ${\lambda }_j$ be invariant means on G. Then $T_{j,{\mu }_j}=T_{j,{\lambda }_j}$.

Lemma 3.4 Assume that an invariant mean exists on G. Then, for each $x\in E$, the set $\overline{co}\{T_{j,t}x:t\in G\}\cap F_j(G)$ consists of a single point.

Proof Let $x\in E$ and ${\mu }_j$ be an invariant mean on G. Then we know that $T_{j,{\mu }_j}x\in F_j(G)$ and $T_{j,{\mu }_j}x\in \overline{co}\{T_{j,t}x:t\in G\}$. So, we show that $\overline{co}\{T_{j,t}x:t\in G\}\cap F_j(G)=\{T_{j,{\mu }_j}x\}$. Let $x_0\in \overline{co}\{T_{j,t}x:t\in G\}\cap F_j(G)$ and $ϵ>0$. Then, for $x^{\ast }\in E^{\ast }$, there exists an element ${\sum }_{i=1}^n{\alpha }_i{T}_{j,t_i}x$ in the set $co\{T_{j,t}x:t\in G\}$ such that $ϵ>C\cdot {\parallel x^{\ast }\parallel }_j\cdot {\parallel {\sum }_{i=1}^n{\alpha }_i{T}_{j,t_i}x-x_0\parallel }_j$. Hence, we have

$$\begin{array}{rcl}ϵ& >& C\cdot {\parallel x^{\ast }\parallel }_j\cdot {\parallel \sum _{i=1}^n{\alpha }_i{T}_{j,t_i}x-x_0\parallel }_j\\ \ge & \underset{t}{sup}{\parallel T_{j,t}\parallel }_j\cdot {\parallel \sum _{i=1}^n{\alpha }_i{T}_{j,t_i}x-x_0\parallel }_j\cdot {\parallel x^{\ast }\parallel }_j\\ \ge & \underset{t}{sup}{\parallel \sum _{i=1}^n{\alpha }_i{T}_{j,t}{T}_{j,t_i}x-x_0\parallel }_j\cdot {\parallel x^{\ast }\parallel }_j\\ \ge & \left|〈\sum _{i=1}^n{\alpha }_i{T}_{j,t}{T}_{j,t_i}x-x_0,x^{\ast }〉\right|\\ =& \left|\sum _{i=1}^n{\alpha }_i{\mu }_{j,t}〈T_{j,t{t}_i}x-x_0,x^{\ast }〉\right|\\ =& \left|{\mu }_{j,t}〈T_{j,t}x-x_0,x^{\ast }〉\right|\\ =& \left|〈T_{j,{\mu }_j}x-x_0,x^{\ast }〉\right|.\end{array}$$

Since ϵ is arbitrary, we have $〈T_{j,{\mu }_j}x,x^{\ast }〉=〈x_0,x^{\ast }〉$ for every $x^{\ast }\in E^{\ast }$ and hence $T_{j,{\mu }_j}x=x_0$. □

4 Ergodic theorems

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

Theorem 4.1 Let $\{T_{j,t}:t\in G\}$ be a family of linear continuous operators in a real Banach space E satisfying Assumption (A). If a net $\{{\mu }_j^{\alpha }:\alpha \in I\}$ of means on G is asymptotically invariant, i.e.,

$${\mu }_j^{\alpha }-r_s^{\ast }{\mu }_j^{\alpha }\mathit{\text{and}}{\mu }_j^{\alpha }-l_s^{\ast }{\mu }_j^{\alpha }$$

converge to 0 in the weak^∗ topology of $m_j{(G)}^{\ast }$ for each $s\in G$, then there exists a projection $Q_j$ of E on to $F_j(G)$ such that ${\parallel (Q_1,\dots ,Q_j)\parallel }_j\le C$, $T_{j,{\mu }_j^{\alpha }}x$ converges weakly to $Q_{j}x$ for each $x\in E$, $Q_j{T}_{j,t}=T_{j,t}{Q}_j=Q_j$ for each $t\in G$, and $Q_{j}x\in \overline{co}\{T_{j,t}x:t\in G\}$ for each $x\in E$. Furthermore, the projection $Q_j$ onto $F_j(G)$ is the same for all asymptotically invariant nets.

Proof Let ${\mu }_j$ be a cluster point of net $\{{\mu }_j^{\alpha }:\alpha \in I\}$ in the weak^∗ topology of $m_j{(G)}^{\ast }$. Then ${\mu }_j$ is an invariant mean on G. Hence, by Lemma 3.2, $T_{j,{\mu }_j}$ is a projection of E onto $F_j(G)$ such that ${\parallel (T_{1,{\mu }_1},\dots ,T_{j,{\mu }_j})\parallel }_j\le C$, $T_{j,{\mu }_j}{T}_{j,t}=T_{j,t}{T}_{j,{\mu }_j}=T_{j,{\mu }_j}$ for each $t\in G$ and $T_{j,{\mu }_j}x\in \overline{co}\{T_{j,t}x:t\in G\}$ for each $x\in E$. Setting $Q_j=T_{j,{\mu }_j}$, we show that $T_{j,{\mu }_j^{\alpha }}x$ converges weakly to $Q_{j}x$ for each $x\in E$. Since $T_{j,{\mu }_j^{\alpha }}x\in \overline{co}\{T_{j,t}x:t\in G\}$ for all $\alpha \in I$ and $\overline{co}\{T_{j,t}x:t\in G\}$ is weakly compact, there exists a subnet $\{T_{j,{\mu }_j^{\beta }}x:\beta \in J\}$ of $\{T_{j,{\mu }_j^{\alpha }}x:\alpha \in I\}$ such that $T_{j,{\mu }_j^{\beta }}x$ converges weakly to an element $x_0\in \overline{co}\{T_{j,t}x:t\in G\}$. To show that $T_{j,{\mu }_j^{\alpha }}x$ converges weakly to $Q_{j}x$, it is sufficient to show $x_0=Q_{j}x$. Let $x^{\ast }\in E^{\ast }$ and $s\in G$. Since $T_{j,{\mu }_j^{\beta }}x\to x_0$ weakly, we have ${\mu }_{j,t}^{\beta }〈T_{j,t}x,x^{\ast }〉\to 〈x_0,x^{\ast }〉$ and ${\mu }_{j,t}^{\beta }〈T_{j,t}x,T_{j,s}^{\ast }{x}^{\ast }〉\to 〈x_0,T_{j,s}^{\ast }{x}^{\ast }〉=〈T_{j,s}{x}_0,x^{\ast }〉$. On the other hand, since ${\mu }_j^{\beta }-l_s^{\ast }{\mu }_j^{\beta }\to 0$ in the weak^∗ topology, we have

$$\begin{array}{c}{\mu }_{j,t}^{\beta }〈T_{j,t}x,x^{\ast }〉-l_s^{\ast }{\mu }_{j,t}^{\beta }〈T_{j,t}x,x^{\ast }〉\hfill \\ ={\mu }_{j,t}^{\beta }〈T_{j,t}x,x^{\ast }〉-{\mu }_{j,t}^{\beta }〈T_{j,st}x,x^{\ast }〉\hfill \\ ={\mu }_{j,t}^{\beta }〈T_{j,t}x,x^{\ast }〉-{\mu }_{j,t}^{\beta }〈T_{j,t}x,T_{j,s}^{\ast }{x}^{\ast }〉\hfill \\ \to 0.\hfill \end{array}$$

Hence, we have $〈x_0,x^{\ast }〉=〈T_{j,s}{x}_0,x^{\ast }〉$ and hence $x_0\in F_j(G)$. So, we obtain $Q_{j}x=T_{j,{\mu }_j}x=x_0$ 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 $\{T_{j,t}:t\in G\}$ 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 ${\parallel (Q_1,\dots ,Q_j)\parallel }_j\le C$, $Q_j{T}_{j,t}=T_{j,t}{Q}_j=Q_j$ for each $t\in G$ and $Q_{j}x\in \overline{co}\{T_{j,t}x:t\in G\}$ for each $x\in E$.

Theorem 4.3 Let $\{T_{j,t}:t\in G\}$ be as in Theorem  4.1. If a net $\{{\mu }_j^{\alpha }:\alpha \in I\}$ of means on G is asymptotically invariant and further ${\mu }_j^{\alpha }-r_s^{\ast }{\mu }_j^{\alpha }$ converges to 0 in the strong topology of $m_j{(G)}^{\ast }$, then there exists a projection $Q_j$ of E onto $F_j(G)$ such that ${\parallel (Q_1,\dots ,Q_j)\parallel }_j\le C$, $T_{j,{\mu }_j^{\alpha }}x$ converges strongly to $Q_{j}x$ for each $x\in E$, $Q_j{T}_{j,t}=T_{j,t}{Q}_j=Q_j$ for each $t\in G$, and $Q_{j}x\in \overline{co}\{T_{j,t}x:t\in G\}$ for each $x\in E$.

Proof As in the proof of Theorem 4.1, let $Q_j=T_{j,{\mu }_j}$, where ${\mu }_j$ is a cluster point of the net $\{{\mu }_j^{\alpha }:\alpha \in I\}$ in the weak^∗ topology of $m_j{(G)}^{\ast }$. We show that $T_{j,{\mu }_j^{\alpha }}x$ converges strongly to $Q_{j}x$ for each $x\in E$.

Let $E_0=\overline{co}\{y-T_{j,t}y:y\in E,t\in G\}$. Then, for any $z\in E_0$, $T_{j,{\mu }_j^{\alpha }}z$ converges strongly to 0. In fact, if $z=y-T_{j,s}y$, then since, for any $y^{\ast }\in E^{\ast }$,

$$\begin{array}{rcl}\left|〈T_{j,{\mu }_j^{\alpha }}z,y^{\ast }〉\right|& =& \left|{\mu }_{j,t}^{\alpha }〈T_{j,t}(y-T_{j,s}y),y^{\ast }〉\right|\\ =& |{\mu }_{j,t}^{\alpha }〈T_{j,t}y,y^{\ast }〉-{\mu }_{j,t}^{\alpha }〈T_{j,ts}y,y^{\ast }〉|\\ =& \left|({\mu }_{j,t}^{\alpha }-r_s^{\ast }{\mu }_{j,t}^{\alpha })〈T_{j,t}y,y^{\ast }〉\right|\\ \le & {\parallel ({\mu }_1^{\alpha }-r_s^{\ast }{\mu }_1^{\alpha },\dots ,{\mu }_j^{\alpha }-r_s^{\ast }{\mu }_j^{\alpha })\parallel }_j\cdot \underset{t}{sup}\left|〈T_{j,t}y,y^{\ast }〉\right|\\ \le & {\parallel ({\mu }_1^{\alpha }-r_s^{\ast }{\mu }_1^{\alpha },\dots ,{\mu }_j^{\alpha }-r_s^{\ast }{\mu }_j^{\alpha })\parallel }_j\cdot C\cdot {\parallel y\parallel }_j\cdot {\parallel y^{\ast }\parallel }_j,\end{array}$$

we have ${\parallel (T_{1,{\mu }_1^{\alpha }}z,\dots ,T_{j,{\mu }_j^{\alpha }}z)\parallel }_j\le C\cdot {\parallel ({\mu }_1^{\alpha }-r_s^{\ast }{\mu }_1^{\alpha },\dots ,{\mu }_j^{\alpha }-r_s^{\ast }{\mu }_j^{\alpha })\parallel }_j\cdot {\parallel y\parallel }_j$. Using this inequality, we show that $T_{j,{\mu }_j^{\alpha }}z$ converges strongly to 0 for any $z\in E_0$. Let z be any element of $E_0$ and ϵ be any positive number. By the definition of $E_0$, there exists an element ${\sum }_{i=1}^n{a}_i(y_i-T_{j,s_i}{y}_i)ϵ$ in the set $co\{y-T_{j,s}y:y\in E,s\in G\}$ such that $ϵ>2C\cdot {\parallel (z-{\sum }_{i=1}^n{a}_i(y_i-T_{1,s_i}{y}_i),\dots ,z-{\sum }_{i=1}^n{a}_i(y_i-T_{j,s_i}{y}_i))\parallel }_j$. On the other hand, from ${\parallel ({\mu }_1^{\alpha }-r_s^{\ast }{\mu }_1^{\alpha },\dots ,{\mu }_j^{\alpha }-r_s^{\ast }{\mu }_j^{\alpha })\parallel }_j\to 0$ for all $s\in G$, there exists $a_0\in I$ such that, for all $\alpha \ge {\alpha }_0$ and $i=1,2,\dots ,n$,

$$ϵ>{\parallel ({\mu }_1^{\alpha }-r_{s_i}^{\ast }{\mu }_1^{\alpha },\dots ,{\mu }_j^{\alpha }-r_{s_i}^{\ast }{\mu }_j^{\alpha })\parallel }_j\cdot 2C{\parallel y_i\parallel }_j.$$

This yields

$$\begin{array}{c}{\parallel (T_{1,{\mu }_1^{\alpha }}z,\dots ,T_{j,{\mu }_j^{\alpha }}z)\parallel }_j\hfill \\ \le \parallel (T_{1,{\mu }_1^{\alpha }}z-T_{1,{\mu }_1^{\alpha }}(\sum _{i=1}^n{a}_i(y_i-T_{1,s_i}{y}_i)),\hfill \\ \dots ,T_{j,{\mu }_j^{\alpha }}z-T_{j,{\mu }_j^{\alpha }}(\sum _{i=1}^n{a}_i(y_i-T_{j,s_i}{y}_i))){\parallel }_j\hfill \\ +{\parallel (T_{1,{\mu }_1^{\alpha }}(\sum _{i=1}^n{a}_i(y_i-T_{1,s_i}{y}_i)),\dots ,T_{j,{\mu }_j^{\alpha }}(\sum _{i=1}^n{a}_i(y_i-T_{j,s_i}{y}_i)))\parallel }_j\hfill \\ \le {\parallel (T_{1,{\mu }_1^{\alpha }},\dots ,T_{j,{\mu }_j^{\alpha }})\parallel }_j\hfill \\ \cdot {\parallel (z-\sum _{i=1}^n{a}_i(y_i-T_{1,s_i}{y}_i),\dots ,z-\sum _{i=1}^n{a}_i(y_i-T_{j,s_i}{y}_i))\parallel }_j\hfill \\ +\sum _{i=1}^n{\parallel (T_{1,{\mu }_j^{\alpha }}(y_i-T_{1,s_i}{y}_i),\dots ,T_{j,{\mu }_j^{\alpha }}(y_i-T_{j,s_i}{y}_i))\parallel }_j\hfill \\ \le C\cdot {\parallel (z-\sum _{i=1}^n{a}_i(y_i-T_{1,s_i}{y}_i),\dots ,z-\sum _{i=1}^n{a}_i(y_i-T_{j,s_i}{y}_i))\parallel }_j\hfill \\ +\underset{i}{sup}{\parallel ({\mu }_1^{\alpha }-r_{s_i}^{\ast }{\mu }_1^{\alpha },\dots ,{\mu }_j^{\alpha }-r_{s_i}^{\ast }{\mu }_j^{\alpha })\parallel }_j\cdot C\cdot {\parallel y_i\parallel }_j\hfill \\ <\frac{ϵ}{2}+\frac{ϵ}{2}=ϵ.\hfill \end{array}$$

Hence, $T_{j,{\mu }_j^{\alpha }}Z$ converges strongly to 0 for each $z\in E_0$.

Next, assume that $x-T_{j,{\mu }_j}x$ for some $x\in E$ is not contained in the set $E_0$. Then, by the Hahn-Banach theorem, there exists a linear continuous functional $y^{\ast }$ such that $〈x-T_{j,{\mu }_j}x,y^{\ast }〉=1$ and $〈z,y^{\ast }〉=0$ for all $z\in E_0$. So since $x-T_{j,t}x\in E_0$ for all $t\in G$, we have

$$〈x-T_{j,{\mu }_j}x,y^{\ast }〉={\mu }_{j,t}〈x-T_{j,t}x,y^{\ast }〉=0.$$

This is a contradiction. Hence, $x-T_{j,{\mu }_j}$ for all $x\in E$ are contained in $E_0$. Therefore we find that $T_{j,{\mu }_j^{\alpha }}x-T_{j,{\mu }_j}x=T_{j,{\mu }_j^{\alpha }}(x-T_{j,{\mu }_j})$ converges strongly to 0 for all $x\in E$. 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 ${\parallel (T_1^n,\dots ,T_j^n)\parallel }_j\le C$ for $n=1,2,\dots$ , 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 ${\parallel (Q_1,\dots ,Q_j)\parallel }_j\le C$, the Cesaro means $S_{j,n}=\frac{1}{n}{\sum }_{k=1}^n{T}_j^{k}x$ converges strongly to $Q_{j}x$ for each $x\in E$, and $T_j{Q}_j=Q_j{T}_j=Q_j$.

Proof Let $x\in E$. Then, since $\{T_j^{n}x:n=1,2,\dots \}=T_j(\{T^{n-1}x:n=1,2,\dots \})\subset T_j(B(0,\parallel x\parallel \cdot (c+1)))$, where $B(x,r)$ means the closed ball with center x and radius r, the weak closure of $\{T_j^{n}x:n=1,2,\dots \}$ is weakly compact. On the other hand, let $G=\{1,2,3,\dots \}$ with the discrete topology and ${\mu }_j^n$ be a mean on G such that ${\mu }_j^n(f_j)={\sum }_{i=1}^n(\frac{1}{n})f_j(i)$ for each $f_j\in m_j(G)$. Then it is obvious that ${\parallel ({\mu }_1^n-r_k^{\ast }{\mu }_1^n,\dots ,{\mu }_j^n-r_k^{\ast }{\mu }_j^n)\parallel }_j\le \frac{2k}{n}\to 0$ for all $k\in G$. So, it follows from Theorem 4.3 that Corollary 4.4 is true. □

If $G=[0,\mathrm{\infty })$ with the natural topology, then we obtain the corresponding result.

Corollary 4.5 Let E be a real Banach space and $\{T_{j,t}:t\in [0,\mathrm{\infty })\}$ 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 ${\parallel (Q_1,\dots ,Q_j)\parallel }_j\le C$, $\frac{1}{T}{\int }_0^{T}T{\int }_{j,t}xdt$ converges strongly to $Q_{j}x$ for each $x\in E$, and $T_{j,t}{Q}_j=Q_j{T}_{j,t}=Q_j$ for each $t\in [0,\mathrm{\infty })$.

Remark 4.6 $\frac{1}{T}{\int }_0^{T}T{\int }_{j,t}xdt$ are weak vector valued integrals with respect to means on $G=[0,\mathrm{\infty })$. As in Section IV of Rodé [4], we can also obtain the strong convergence of the sequences

$$(1-r)\sum _{k=1}^{\mathrm{\infty }}{r}^k{T}_j^{k}x,r\to 1-$$

and

$$\lambda {\int }_0^{\mathrm{\infty }}{e}^{-\lambda t}{T}_{j,t}xdt,\lambda \to 0+.$$