Cesàro and Abel ergodic theorems for integrated semigroups

: Let { S ( t ) } t ≥0 be an integrated semigroup of bounded linear operators on the Banach space X into itself and let A be their generator. In this paper, we consider some necessary and sufficient conditions for the Cesàro mean and the Abel average of S ( t ) converge uniformly on B ( X ). More precisely, we show that the Abel average of S ( t ) converges uniformly if and only if X = R ( A ) ⊕ N ( A ), if and only if R ( A k ) is closed for some integer k and (cid:107) λ 2 R ( λ , A ) (cid:107) −→ 0 as λ → 0 + , where R ( A ), N ( A ) and R ( λ , A ), be the range, the kernel, the resolvent function of A , respectively. Furthermore, we prove that if S ( t )/ t 2 −→ 0 as t → ∞, then the Cesàro mean of S ( t ) converges uniformly if and only if the Abel average of S ( t ) is also converges uniformly.


Introduction
Throughout this paper B(X) denotes the Banach algebra of all bounded linear operators on Banach space X into itself. Let A be a closed linear operator on X with domain D(A) ⊂ X, we denote by N(A), R(A), σ(A), ρ(A) and R(., A), the kernel, the range, the spectrum, the resolvent set and the resolvent operator of A, respectively.
The family {T(t)} t≥ of bounded linear operator on X is called a strongly continuous semigroup (Csemigroup in short [2]) if it has the following properties:

T( ) = I, 2. T(t)T(s) = T(t + s), 3. The map t → T(t)x from [ , +∞[ into X is continuous for all x ∈ X.
Their in nitesimal generator A is de ned by: which is exactly the resolvent function of A. Moreover, the in nitesimal generator of a C -semigroup is a linear closed densely de ned operator on a Banach space X, see for instance [12] and [3]. Integrated semigroups and n-time Integrated semigroups, n ∈ N, of operators in Banach space were introduced by Arendt [1] and studied by Arendt, Kellermann, Hieber [7], Thieme [17] and many others. A relevant example is obtained if we assume that {T(t)} t is a C -semigroup of bounded linear operator on Arendt [1] showed that certain natural classes of operators, such as adjoint semigroups of C -semigroups on non-re exive Banach spaces, give rise to integrated semigroups which are not integrals of C -semigroups.
In contrast to C -semigroups, integrated semigroups may be not exponentially bounded, may be locally de-ned, and their generators may be not densely de ned.
Ergodic theorems have a long tradition and are usually formulated in semigroup theory via the existence of limits of the Cesàro means, de ned by: where {T(t)} t≥ ⊂ B(X) is a C -semigroup. In this case, {T(t)} t≥ is said to be uniformly Cesàro ergodic if the Cesàro means of T(t) converges uniformly in B(X), as t → ∞. This notion is completely connected to study the limit of the Abel averages of a C -semigroup {T(t)} t≥ , de ned as: where A is the generator of {T(t)} t≥ and R(λ, A) is the resolvent function of A.
Recall that, a C -semigroup T(t) is called uniformly Abel ergodic if the limit, as λ → + , if their Abel averages exists in the norm operator topology. We will denote the growth bound of a C -semigroup {T(t)} t≥ by Usually one assumes ω ≤ or the even stronger condition T(t) /t −→ as t → ∞, to study the convergence of the Cesàro means and the Abel averages of {T(t)} t≥ . Generally, great attention has been focused on the study the relationship between Cesàro ergodicity and Abel ergodicity for di erent classes of semigroups in B(X). The result of Hille and Phillips in [6,Theorem 18.8.4] deals with the uniform Abel ergodicity of semigroups of class (A), a class slightly larger than C -semigroups, under the assumption ω ≤ . More precisely, they have shown that T(t) is uniformly Abel ergodic if and only if λ R(λ, A)x −→ as λ → + for every x ∈ X and X = R(A) ⊕ N(A). Furthermore, if T(t) is uniformly Abel ergodic, then R(A m ) = R(A), for all m ∈ N * . A relevant result obtained by S.Y. Shaw in [13] for a locally integrated semigroup, under an assumption weaker than ω ≤ , that means T(t) is uniformly Cesàro ergodic if and only if it satis es the following conditions: (i) The Laplace transformation R λ exists for every λ > , (ii) T(t)R λ /t −→ as t → ∞, for some λ > , and (iii) T(t) is uniformly Abel ergodic. The condition (i) holds whenever ω ≤ , let us also mention that somewhat di erent necessary and su cient conditions are obtained in [14]. Clearly, if T(t) is uniformly Cesàro ergodic then is uniformly Abel ergodic, but the reverse is not true, for more information see [9,Chapter 2]. It is useful to mention that the limit of Cesàro averages and of Abel averages of C -semigroup {T(t)} t is the same, is the projection P of X onto N(A) parallel to R(A), corresponding to the ergodic decomposition (1. 3) The classical uniform ergodic theorem for C -semigroups {T(t)} t≥ of bounded linear operators on X, goes back to M. Lin in [10], he treats the Cesàro ergodicity of a C -semigroup {T(t)} t≥ under the assumption lim Lin, Shoikhet and Suciu in [11], showed that for a C -semigroup {T(t)} t≥ on B(X) satisfying lim t→∞ T(t) /t = , {T(t)} t≥ is uniformly ergodic if and only if there exists some λ > such that the Abel average λR(λ, A) is uniformly power convergent. Kozitsky, Shoikhet and Zemànek in [8], obtained necessary and su cient conditions for which the Abel averages of a C -semigroup T(t) can be uniformly power convergent. Further condition have been obtained more recently by several authors [8,16]. This paper is organized as follows. In section 2, we give some de nitions and fundamental properties for an integrated semigroup of bounded linear operators on X.

Preliminaries
We start this present section by recalling an interesting concept in operator theory that we need in the sequel. Let The di erentiation spaces C n , n ≥ , are de ned by C = X and Using this notion (2) can equivalently be formulated by S(t)x ∈ C for all x ∈ X, and Usually, a non-degenerate integrated semigroup is uniquely determined by its generator. Motivated from the Laplace transform theory we can de ne the generator of an integrated semigroup as In this case, the Laplace Transformation R λ exists for all λ with Reλ > ω. In General, an operator A is called generator of an integrated semigroup, if there exists ω ∈ R such that (ω, ∞) ⊂ ρ(A), and there exists a strongly continuous exponentially bounded family {S(t)} t of bounded linear operators on X such that In this case, the Laplace Transformation R λ of {S(t)} t is exactly the resolvent function of A and satis es the pseudo-resolvent: For more details, we refer the interested reader to [1,5,7].
The integrated semigroup generated by A is given by 2. We consider X = and the family {S(t)} t of bounded linear operators on X de ned by: where an = n + n πi. Then {S(t)} t is an integrated semigroup on X.

Consider the Schrödinger operator A = i.∆ on L p (R) for p ≥ . Then A generates an integrated semigroup
({S(t)} t≥ given by converges in the norm operator topology as t tend to in nity. Moreover, (

ii) We say that {S(t)} t is uniformly Abel ergodic if the Abel averages of {S(t)} t de ned by
converges in the norm operator topology as λ tend to zero.
Now, we will need the following relations between integrated semigroups and their generators. The rst result was investigated by W. Arendt [1] in the case of n-times integrated semigroup on B(X), where n ∈ N.
We recall the following result which was recently proved in [16] for an α-times integrated semigroup

Main results
We began this section by the following two lemmas which will be widely used in the sequel.

Lemma 3.1. Let A be the generator of an integrated semigroup
Proof. Let {S(t)} t be an integrated semigroup on B(X) with A be their generator and R(λ, A) the resolvent function of A. We assume that λ R(λ, A) → as λ → + and let y ∈ R(A) ∩ N(A), then by the second assertion of Lemma 2.1, we get λR(λ, A)y = y , for all λ ∈ ρ(A).
Since R(A) = R λR(λ, A) − I , then there exists x ∈ X and M > , such that Next, from the resolvent equation: We get the following inequality, for all λ ≠ µ, It follows that λR(λ, A)y −→ as λ → + , which yields that y = . Therefore R(A) ∩ N(A) = { }. Then, for a linear operator A on a Banach space X and for some integers j and k, we get Then, we obtain the following results: Next, we assume that R(A n ) + N(A m ) is closed for some n > d and m ≥ . It follows from (ii) that R(A n ) is closed for some n > d. Again, we applied the rst argument, we deduce R(A n ) is closed for all n ≥ d, so is R(A n ) + N(A m ) for all n ≥ d and m ≥ .
The rst main result of this paper can be described as follows.

Theorem 3.1. Let A be the generator of an integrated semigroup {S(t)} t on B(X).
Then the following assertions are equivalent:

S(t) is uniform
Then y ∈ N λR(λ, A) − I , which implies that y = . Thus A is one to one. Clearly, we have R(λ, A)Y ⊂ Y, hence we obtain that λR(λ, A) − I Y ⊂ R(A ). Then, we get the following Hence Y = R(A ), so A − is de ned on all Y, since A is closed, therefore A − is closed, and by the closed graph theorem A − is continuous.

Corollary 3.1. Let A be the generator of an integrated semigroup {S(t)} t on B(X).
Then the following assertions are equivalent:  It follows that, we get Therefore the above estimate implies that λR(λ, A)− P −→ as λ → + , which yields that S(t) is uniformly Abel ergodic and the proof is nished. Using the resolvent equation, we obtain for all x ∈ X and µ ≠ λ; From the identity (3.5), we obtain It follows from the above estimate that λ R(λ, A) −→ as λ → + . So, according to Corollary 3.1, Theorem 3.1 and Theorem 3.2, the equivalents hold.    So, we obtain Then C(t)x − Px −→ as t → ∞. Finally, S(t) is uniformly Cesàro ergodic. A is one to one by [15,Corollary 2.4]. In this case, if the Cesàro averages and the Abel averages converge, they will converge to zero and we obtain X = R(A). Moreover, the strong limit of t t S(r)dr may be divergent when t → ∞, as the following example shows. is not dense in X, then A cannot be an in nitesimal generator of a C -semigroup. Furthermore, the semigroup S(t) generated by A, is given by: x f (s)ds, si ≤ x ≤ t, .
Note that S(t) is an integrated semigroup of type ω = , where ω = inf{w ∈ R : there exists M such that S(t) ≤ Me wt , t ≥ }.
It follows that S(t) /t −→ as t → ∞. On the other hand, since the generator A has an empty spectrum, then X = R(A), and by Theorem 3.1 and Corollary 3.3, we deduce that S(t) is both uniformly Cesàro ergodic and uniformly Abel ergodic. More precisely, we have lim t→∞ t t S(r)dr = lim λ→ + λ ∞ e −λt S(t)dt = .
Now, we suppose that then there exists an operator P such that lim t→∞ t t S(r)dr − P = .
Hence P is a projection operator of X onto N(A) parallel to R(A), corresponding to ergodic decomposition X = R(A) ⊕ N(P). As shown above that X = R(A), then we deduce that lim t→∞ t t S(r)dr = .
Next, let f be a non-zero function de ned on X, hence there exists g ∈ D(A) such that f = Ag = −g . Therefore, we obtain for < x ≤ s By assumption lim t→∞ t t S(r)dr = , hence g(x) = for all < x ≤ s, absurd.
Finally, t t S(r)dr is divergent as t tends to in nity.