2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 227–229. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

UNIFORMLY ERGODIC THEOREM FOR COMMUTING MULTIOPERATORS

SAMIR LAHRECH, ABDERRAHIM MBARKI, ABDELMALEK OUAHAB, SAID RAIS

Abstract. In this paper, we established some uniformly Ergodic theorems by using multioperators satisfying the E-k condition introduce in [3]. One consequence, is that if I −T is quasi-Fredholm and satisfies E-k condition then T is uniformly ergodic. Also we give some conditions for uniform ergodicity of a commuting multioperators satisfies condition E-k. These results are of interest in view of analogous results for unvalued operators (see, for example [2]) also in view of the recent developments in the ergodic theory and its applications.

1. Introduction and main results

Throughout this paper, X is a complex Banach space, and L(X) is the algebra of linear continuous operators acting in X. If there is an integer n for which T^{n+1}X = T^{n}X, then we say that T has finite descent and the smallest integer d(T) for which equality occurs is called the descent of T. If there is exists an integer m for which kerT^{m+1} = kerT^{m}, then T is said to have finite ascent and the smallest integer a(T) for this equality occurs is called ascent of T. If both a(T) and d(T) are finite, then they are equal [1, 38.3]. We say that T is chain-finite and that its chain length is this common minimal value. Moreover [1, 38.4], in this case there is a decomposition of the vector space

X = T^{d(T)}X ⊕ ker T^{d(T)}.

We now focus on the topological situation: For every T ∈ L(X) we set

M_{i}(T) = i^{−1}(I + T + T^{2} + ⋯ + T^{i−1}), i = 1, 2, 3, …,  (1.1)

i.e. the averages associated with T, where I = id_{X} is the identity of X. If T = (T_{1}, T_{2}, …, T_{n}) ∈ L(X)^{n} is commuting multioperator (briefly, c.m.), we also set

M_{v}(T) = M_{v_{1}}(T_{1})M_{v_{2}}(T_{2}) … M_{v_{n}}(T_{n}), v ∈ Z_{+}^{n}, v ≥ e,  (1.2)

where Z_{+}^{n} is the family of multi-indices of length n (i.e. n-tuples of nonnegative integers) and e := (1, 1, …, 1) ∈ Z_{+}^{n}. In other words, (1.2) defines the averages associated with T.

2000 Mathematics Subject Classification. 47A35, 47A13.

Key words and phrases. Average; E-k condition; finite descent; uniform ergodicity.

\begin{matrix}c \\ ◯ \end{matrix}2006 Texas State University - San Marcos.

Published September 20, 2006.

Definition 1.1. A commuting multioperator T ∈ L(X)^{n} is said to be uniformly ergodic if the limit

lim_{v} M_{v}(T)  (1.3)

exists in the uniform topology of L(X).

Remark 1.2. (a) If n = 1, then (1.3) is automatically fulfilled, and therefore the above definition extends the usual concept of uniformly ergodic operator (see, for example [2]).

(b) If T = (I, …, T_{j}, I, …, I) ∈ L(X)^{n}, then T is uniformly ergodic if and only the lim_{v_{j}} M_{v_{j}}(T_{j}) exists in the uniform topology of L(X).

Definition 1.3 ([3]). Let k = (k_{1}, … k_{n}) ∈ Z_{+}^{n} and T ∈ L(X)^{n} be a c.m. We say that T satisfies condition E-k if lim_{v}(I − T_{j})^{k_{j}}M_{v}(T) = 0 for each j ∈ {1, …, n}.

It is clear that condition E-k implies condition E-n for any n ≥ k Thus we see that the example T = (T_{1}, I, …, I) ∈ Z_{+}^{n} with

T_{1} = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) .

This shows that E-2e is strictly weaker than E-e.

Theorem 1.4. Let k ∈ Z_{+}^{n}. Suppose T ∈ L(X)^{n} satisfies condition E-k and ∑_{j=1}^{n}(I − T_{j})^{k_{j}}X, ∑_{j=1}^{n}(I^{∗} − T_{j}^{∗})^{k_{j}}X^{∗} are closed in X and X^{∗} respectively. If [ ∑_{j=1}^{n}(I − T_{j})^{k_{j}}X] ∩ [ ∩_{j=1}^{n} ker(I − T_{j})^{k_{j}}] = {0}. Then T is uniformly ergodic

Proof. Arguing exactly as in [5, Theorem 1], with δ_{T} and γ_{T} given by

⊕_{j=1}^{n}x_{j} → δ_{T} (⊕_{j=1}^{n}x_{j}) = ∑_{j=1}^{n}(I − T_{j})^{k_{j}}x_{j} and x → γ_{T} (x) = ⊕_{j=1}^{n}(I − T_{j})^{k_{j}}x.

□

Theorem 1.5. Let T ∈ L(X) satisfy condition E-r, and one of the following nine conditions:

(a) I − T has chain length at most r

(b) 1 is a pole of the resolvent of order at most r

(c) I − T is quasi-Fredholm operator

(d) (I − T)^{r}X is closed and ker(I − T)^{r} has a closed T -invariant complement (e) (I − T)^{r}X ⨁ ker(I − T)^{r} = (I − T)^{r}X + ker(I − T)^{r}

(f) (I − T)^{m}X is closed for all m ≥ r

(g) (I − T)^{r}X is closed

(h) (I − T)^{m}X is closed for some m ≥ r

(i) I − T has finite descent. Then T is uniformly ergodic.

Proof. Firstly, from [3, Theorem 6], the above statements (a)–(i) are equivalent. Then, take G = (T, I, … I) ∈ L(X)^{n} and k = (r, 1, …, 1) ∈ Z_{+}^{n}; Therefore, we have ∑_{j=1}^{n}(I −G_{j})^{k_{j}}X = (I −T)^{r}X is closed, it follows that ∑_{j=1}^{n}(I^{∗} −G_{j}^{∗})^{k_{j}}X^{∗} = (I^{∗} − T^{∗})^{r}X^{∗} is closed, which implies since (I − T)^{r}X ∩ ker(I − T)^{r} = {0}, that G is uniformly ergodic in L(X)^{n}. From Theorem 1.4 this means that T is uniformly ergodic in L(X) □

Theorem 1.6. Let k ∈ Z_{+}^{n}. If T ∈ L(X)^{n} A c.m. satisfies condition E-k, such that ∑_{j=1}^{n}(I −T_{j}) has chain length at most 1 and ker(∑_{j=1}^{n}(I −T_{j})) = ∩_{j=1}^{n} ker(I −T_{j}). Then T is uniformly.

Proof. There are two cases Case 1: d( ∑_{j=1}^{n}(I −T_{j})) = 0. Then ∑_{j=1}^{n}(I −T_{j}) is bijective then X = ∑_{j=1}^{n}(I − T_{j})X ⊕ ker(∑_{j=1}^{n}(I − T_{j})), which implies, since ∩_{j=1}^{n} ker(I − T_{j}) ⊂ ker(∑_{j=1}^{n}(I − T_{j})) = {0} that X = ∑_{j=1}^{n}(I − T_{j})X ⨁ ∩_{j=1}^{n} ker(I − T_{j}), from the [3, Theorem 10] we obtain T is uniformly ergodic. Case 2: d(( ∑_{j=1}^{n}(I −T_{j}))) = 1. Then ( ∑_{j=1}^{n}(I −T_{j}))X = ( ∑_{j=1}^{n}(I −T_{j}))^{2}X, so ( ∑_{j=1}^{n}(I−T_{j}))X = ( ∑_{j=1}^{n}(I−T_{j}))^{nr}( ∑_{j=1}^{n}(I−T_{j}))X, with r = max_{1≤j≤n} k_{j}. so ( ∑_{j=1}^{n}(I−T_{j}))^{nr} is a bijection of ( ∑_{j=1}^{n}(I−T_{j}))X onto itself. Which implies, since T satisfies condition E-k, that M_{v}(T)|( ∑_{j=1}^{n}(I −T_{j}))X → 0 and since M_{v}(T)|∩_{j=1}^{n} ker(I − T_{j}) = I| ∩_{j=1}^{n} ker(I − T_{j}), it follows that T is uniformly ergodic. □

References

[1] Heuser H. G., Functional analysis, Wiley, Chichester, 1982.

[2] Krengel U., Ergodic theorems, De Gruyter, Berlin and New York, 1985.

[3] Lahrech S., A. Azizi, A. Mbarki, A. Ouahab; Uniformly Ergodic theorem and finite chain length for multioperators International Journal of Pure and Applied Mathematics, 22, No. 2 (2005), 167-172.

[4] Laursen K. B. and Mbekhta M.; Operators with finite chain length and the ergodic theorem, Proc. Amer. Math. Soc. 123 (1995), 3443-3448.

[5] Mbekhta M. and Vasilescu F.-H., Uniformly ergodic multioperators, Trans. Amer. Math. Soc. 347, (1995), 1847- 1854.

Samir Lahrech Département de Mathématiques, Université Oujda, 60000 Oujda, Morocco

   E-mail address: lahrech@sciences.univ-oujda.ac.ma

Abderrahim Mbarki Current address: National school of Applied Sciences, P.O. Box 669, Oujda University, Morocco

   E-mail address: ambarki@ensa.univ-oujda.ac.ma

Abdelmalek Ouahab Département de Mathématiques, Université Oujda, 60000 Oujda, Morocco

   E-mail address: ouahab@sciences.univ-oujda.ac.ma

Said Rais Département de Mathématiques, Université Oujda, 60000 Oujda, Morocco

   E-mail address: said rais@yahoo.fr