C_0$$ C 0 -semigroups associated with uniquely ergodic Kantorovich modifications of operators (original) (raw)

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.

On quasi-contractivity of C0-semigroups on Banach spaces

Archiv der Mathematik, 2004

A basic result in semigroup theory states that every C 0-semigroup is quasi-contractive with respect to some appropriately chosen equivalent norm. This paper contains a counterpart of this well-known fact. Namely, by examining the convergence of the Trotter-type formula (e t n A P) n (where P denotes a bounded projection), we prove that whenever the generator A is unbounded it is possible to introduce an equivalent norm on the space with respect to which the semigroup is not quasi-contractive. Mathematics subject classification (2000): 47A05, 47D06

Iterates of multidimensional Kantorovich- type operators and their associated positive C0-semigroups

2011

In this paper we deepen the study of a sequence of positive linear operators acting on L([0, 1] ), N ≥ 1, that have been introduced in [3] and that generalize the multidimensional Kantorovich operators (see [15]). We show that particular iterates of these operators converge on C ([0, 1] ) to a Markov semigroup and on L([0, 1] ), 1 ≤ p < +∞, to a positive contractive C0-semigroup (that is an extension of the previous one). The generators of these C0-semigroups are the closures of some partial differential operators that belong to the class of Fleming-Viot operators arising in population genetics. Mathematics Subject Classification (2010): 41A36, 47D06, 47F05.

On a class of positive C 0-semigroups of operators on weighted continuous function spaces

2011

This paper is mainly concerned with the study of the generators of those positive C0-semigroups on weighted continuous function spaces that leave invariant a given closed sub-lattice of bounded continuous functions and whose relevant restrictions are Feller semigroups. Additive and multiplicative perturbation results for this class of generators are also established. Finally, some applications concerning multiplicative perturbations of the Laplacian on R n , n ≥ 1, and degenerate second-order differential operators on unbounded real intervals are showed.

On generators of C0-semigroups of composition operators

Israel Journal of Mathematics, 2018

that an (unbounded) operator (Af ) = G • f ′ on the classical Hardy space generates a C 0 semigroup of composition operators if and only if it generates a quasicontractive semigroup. Here we prove that if such an operator A generates a C 0 semigroup, then it is automatically a semigroup of composition operators, so that the condition of quasicontractivity of the semigroup in the cited result is not necessary. Our result applies to a rather general class of Banach spaces of analytic functions in the unit disc. 2010 Mathematics Subject Classification. 47B35 (primary). Key words and phrases. C 0 -semigroup of composition operators.