Ergodicity for a Family of Operators

: The aim of this paper is to introduce the notions of power boundedness, Cesàro boundedness, mean ergodicity, and uniform ergodicity for a family of bounded linear operators on a Banach space. The authors present some elementary results in this setting and show that some main results about power bounded, Cesàro bounded, mean ergodic, and the uniform ergodic operator can be extended from the case of a linear bounded operator to the case of a family of bounded linear operators acting on a Banach space. Also, we show that the Yosida theorem can be extended from the case of a bounded linear operator to the case of a family of bounded linear operators acting on a Banach space.


Introduction
Let T be a bounded linear operator, on a complex Banach space X.The ergodicity for T was already developed in different directions, (see, e.g.[2,3,4,6,7,8,9,12]).For example, in [3], it was shown that when 1  n T n −→ 0, n → ∞, T is uniformly ergodic if and only if (I − T ) 2 X is closed.In [9], Lin showed that if 1  n T n −→ 0, n → ∞, T is uniformly ergodic if and only if (I − T )X is closed.Hence (I − T ) k X is closed for each integer k ≥ 1.In [1], the authors extended this result to the case of a family of bounded linear operators.It is well known that the Cesàro boundedness of the operator T , as well as the condition, lim n−→∞ 1 n T n x = 0, for every x ∈ X, which are, in general, independent; see [5,Remark 4], are necessary for the mean ergodicity of T .In [13], Yosida has established the following theorem for an operator acting on a locally convex space.Theorem 1.1.Let X be a locally convex linear topological space, and T a continuous linear operator on X into X.We assume that the family of operators {T n : n = 1, 2, . ..} is equi-continuous in the sense that, for any continuous semi-norm q on X, there exists a continuous semi-norm q ′ on X such that In particular, In this paper we introduce the notions of power boundedness, Cesàro boundedness, mean ergodicity, and uniform ergodicity for a family of bounded linear operators from the Banach algebra C b ((0, 1], B(X)) (respectively from B ∞ ), this class was introduced and studied by S. Macovei in [10,11].The notion of uniform ergodicity was introduced in [1], see below for the definitions.We deal with giving relations between these definitions.In Theorem 3.10 we show that the restriction of a mean (respectively uniformly) ergodic family of bounded linear operators on an invariant subspace is also mean (respectively uniformly) 2 A. Akrym, A. EL Bakkali and A. Faouzi ergodic.Also, we extend the Theorem 1.1 for a family of bounded linear operators acting on a Banach space, see Theorem 3.13 below.
Papers dedicated to the study of the class of a family of bounded linear operators acting on a Banach space have been elaborated in [10,11].

Preliminaries
Let X be an infinite-dimensional Banach space and B(X) the Banach algebra of all bounded linear operators on X.We denote by I the identity operator on X.
In [10], Macovei showed that the set In [11], Macovei showed that the set is a Banach space in rapport with norm The quotient space X b ((0, 1], X) /X 0 ((0, 1], X), which will be denoted X ∞ , is a Banach space in rapport with quotient norm In [11], it has shown that B ∞ ⊂ B(X ∞ ), where B(X ∞ ) is the algebra of linear bounded operators on X ∞ .
Let T ∈ B(X), we denote the Cesàro means by We say that the operator T is mean ergodic if there exists P ∈ B(X) such that We say that the operator T is uniformly ergodic if there exists P ∈ B(X) such that In the following definitions, we introduce the notions of power boundedness , Cesàro boundedness, mean ergodicity and uniform ergodicity for family of operators of C b ((0, 1], B(X)).
Definition 2.1.We say that a family of operators We say that it is Cesàro bounded if We say that it is uniformly ergodic if there exists . Then, T is mean (respectively uniformly) ergodic if and only if {T h } h∈(0,1] is mean (respectively uniformly) ergodic.
For every integer j define the operator T j by T j (x) = (y n ) n=1,2.... where y n = x n+1 for n ≤ j − 1; y j = 0; y n = x n for n > j.The operator T j acts as a shift to the left on the j first values of x, but acts as the identity on the infinite part of x after j.
One can show that each T j is mean ergodic with the corresponding projector P j defined by P j (x) = (z n ) n=1,2... , where z n = 0 for n ≤ j; z n = x n for n > j.By taking n < j and the constant sequence Now for h ∈ (0, 1] put S h = T j when 1 j+1 < h ≤ 1 j .Each S h is mean ergodic with the projector P h = P j for 1 j+1 < h ≤ 1 j .But with the constant sequence 1, from the property of the operators T j , we have lim sup h→0 M n (S h ) 1 − P h 1 ∞ = 1 for every integer n, and the condition of Definition 2.3 does not hold.
In the following definitions, we introduce the notions of power boundedness, Cesàro boundedness, mean ergodicity, and uniform ergodicity for a family of operators of B ∞ .

Definition 2.6. We say that {T
where We say that it is uniformly ergodic if there exists

Main results
We start this section with some propositions relating power boundedness, Cesàro boundedness, mean ergodicity, and uniform ergodicity of a family of bounded linear operators {T h } ∈ B ∞ .Proposition 3.1.Let {T h } ∈ B ∞ be power bounded.Then any {S h } h∈(0,1] ∈ {T h } is also power bounded.
In particular, we obtain the following results.
In particular, we obtain the following results.
In particular, we obtain the following results.
It is easy to show that Propositions 3.8 and 3.9 below holds.