Stability of operators and operator semigroups (original) (raw)

Mean ergodic semigroups of operators

Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie a-Matematicas, 2012

We present criteria for determining mean ergodicity of C 0-semigroups of linear operators in a sequentially complete, locally convex Hausdorff space X. A characterization of reflexivity of certain spaces X with a basis via mean ergodicity of equicontinuous C 0-semigroups acting in X is also presented. Special results become available in Grothendieck spaces with the Dunford-Pettis property. Keywords C 0-semigroup • (reflexive) locally convex space • (uniform) mean ergodic Mathematics Subject Classification (2000) MSC 46A04 • 47A35 • 47D03 • 46A11

semigroups and mean ergodic operators in a class of Fréchet spaces

Journal of Mathematical Analysis and Applications, 2010

It is shown that the generator of every exponentially equicontinuous, uniformly continuous C0-semigroup of operators in the class of quojection Fréchet spaces X (which includes properly all countable products of Banach spaces) is necessarily everywhere defined and continuous. If, in addition, X is a Grothendieck space with the Dunford-Pettis property, then uniform continuity can be relaxed to strong continuity. Two results, one of M. Lin and one of H.P. Lotz, both concerned with uniformly mean ergodic operators in Banach spaces, are also extended to the class of Fréchet spaces mentioned above. They fail to hold for arbitrary Fréchet spaces. Dedicated to the memory of V. B. Moscatelli 1. Introduction. Consider a C 0-semigroup of operators (T (t)) t≥0 acting in a Banach space X and which is operator norm continuous. It is a classical result that its infinitesimal generator is then an everywhere defined, bounded linear operator on X, [17, Chap. VIII, Corollary 1.9]. If X happens to be a Grothendieck space with the Dunford-Pettis property (briefly, a GDP-space), then the operator norm continuity of (T (t)) t≥0 is automatic whenever the semigroup is merely strongly continuous. This is an elegant result due to H. P. Lotz, [26, 27], which had well known forrunners for particular GDP-spaces and C 0-semigroups of operators. For instance, it was known that every strongly continuous semigroup of positive operators in L ∞ has a bounded generator, [20]. Or, by a result of L.A. Rubel (see [7], for example), given any strongly continuous group of isometries (T (t)) t∈R in H ∞ (D) there exists α ∈ R such that T (t) = e iαt I, for t ∈ R. Hence, T (•) is surely uniformly continuous. Let T be a bounded linear operator on a Banach space X and consider its Cesàro means T [n] := 1 n n m=1 T m , n ∈ N. If the sequence {T [n] } ∞ n=1 converges to some operator strongly in X (resp. in operator norm), then T is called mean ergodic (resp. uniformly mean ergodic). As a standard reference on this topic we refer to [24], for example. A useful result of M. Lin states that if Ker(I − T) = {0} and lim n→∞ 1 n T n = 0, then T is

Uniform mean ergodicity of C_0C_0C_0-semigroups\newline in a class of Fréchet spaces

Functiones et Approximatio Commentarii Mathematici

Let (T (t)) t 0 be a strongly continuous C 0-semigroup of bounded linear operators on a Banach space X such that limt→∞ T (t)/t = 0. Characterizations of when (T (t)) t 0 is uniformly mean ergodic, i.e., of when its Cesàro means r −1 r 0 T (s) ds converge in operator norm as r → ∞, are known. For instance, this is so if and only if the infinitesimal generator A has closed range in X if and only if lim λ↓0 + λR(λ, A) exists in the operator norm topology (where R(λ, A) is the resolvent operator of A at λ). These characterizations, and others, are shown to remain valid in the class of quojection Fréchet spaces, which includes all Banach spaces, countable products of Banach spaces, and many more. It is shown that the extension fails to hold for all Fréchet spaces. Applications of the results to concrete examples of C 0-semigroups in particular Fréchet function and sequence spaces are presented.

Uniform mean ergodicity of C_0-semigroups in a class of Fréchet spaces

2014

Let (T (t)) t≥0 be a strongly continuous C0semigroup of bounded linear operators on a Banach space X such that limt→∞ T (t)/t = 0. Characterizations of when (T (t)) t≥0 is uniformly mean ergodic, i.e., of when its Cesàro means r −1 r 0 T (s) ds converge in operator norm as r → ∞, are known. For instance, this is so if and only if the innitesimal generator A has closed range in X if and only if lim λ↓0 + λR(λ, A) exists in the operator norm topology (where R(λ, A) is the resolvent operator of A at λ). These characterizations, and others, are shown to remain valid in the class of quojection Fréchet spaces, which includes all Banach spaces, countable products of Banach spaces, and many more. It is shown that the extension fails to hold for all Fréchet spaces. Applications of the results to concrete examples of C0semigroups in particular Fréchet function and sequence spaces are presented. 1 N N n=1 R n 1 exists for the operator norm topology in L(X). (4) There exists a projection P ∈ L(X) with ImP = {x ∈ X : T (t)x = x ∀t ≥ 0} such that lim λ↓0 + λR λ − P = 0. (5) lim n→∞ (λR λ) n exists for the operator norm topology in L(X) for (some) all λ > 0.

Ergodic theorems in Banach ideals of compact operators

Sibirskie Elektronnye Matematicheskie Izvestiya, 2021

Let H be an infinite-dimensional Hilbert space, and let B(H) (K(H)) be the C *-algebra of bounded (respectively, compact) linear operators in H. Let (E, • E) be a fully symmetric sequence space. If {sn(x)} ∞ n=1 are the singular values of x ∈ K(H), let C E = {x ∈ K(H) : {sn(x)} ∈ E} with x C E = {sn(x)} E , x ∈ C E , be the Banach ideal of compact operators generated by E. We show that the averages An(T)(x) = 1 n+1 n k=0 T k (x) converge uniformly in C E for any Dunford-Schwartz operator T and x ∈ C E. Besides, if x ∈ B(H) \ K(H), there exists a Dunford-Schwartz operator T such that the sequence {An(T)(x)} does not converge uniformly. We also show that the averages An(T) converge strongly in (C E , • C E) if and only if E is separable and E = l 1 , as sets.

Montel resolvents and uniformly mean ergodic semigroups of linear operators

Quaestiones Mathematicae, 2013

For C0semigroups of continuous linear operators acting in a Banach space criteria are available which are equivalent to uniform mean ergodicity of the semigroup, meaning the existence of the limit (in the operator norm) of the Cesàro or Abel averages of the semigroup. Best known, perhaps, are criteria due to Lin, in terms of the range of the innitesimal generator A being a closed subspace or, whether 0 belongs to the resolvent set of A or is a simple pole of the resolvent map λ → (λ − A) −1. It is shown in the setting of locally convex spaces (even in Fréchet spaces), that neither of these criteria remain equivalent to uniform ergodicity of the semigroup (i.e., the averages should now converge for the topology of uniform convergence on bounded sets). Our aim is to exhibit new results dealing with uniform mean ergodicity of C0semigroups in more general spaces. A characterization of when a complete, barrelled space with a basis is Montel, in terms of uniform mean ergodicity of certain C0semigroups acting in the space, is also presented.

On Ergodic Operator Means in Banach Spaces

Integral Equations and Operator Theory, 2016

We consider a large class of operator means and prove that a number of ergodic theorems, as well as growth estimates known for particular cases, continue to hold in the general context under fairly mild regularity conditions. The methods developed in the paper not only yield a new approach based on a general point of view, but also lead to results that are new, even in the context of the classical Cesàro means.

Cesàro and Abel ergodic theorems for integrated semigroups

Concrete Operators, 2021

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(Ak) is closed for some integer k and ∥ λ 2 R(λ, A) ∥ → 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 → 1, then the Cesàro mean of S(t) converges uniformly if and only if the Abel average of S(t) is also converges uniformly.

UNIFORMLY ERGODIC THEOREM FOR SEMIGROUPS WITH k-DECOMPOSABLE KATO INFINITESIMAL GENERATORS

2005

In this paper we shall extend the technical assumption (E − k) to semigroups. We prove that if T = (T (t), t ≥ 0) is C 0 -semigroup of operators in L(X) with k-decomposable Kato infinitesimal generator A satisfying the condition (E − k), then T is uniformly ergodic. These results are of interest in view of recent activity in the ergodic theory and its applications.