Semiflows and semigroups

Edoardo Vesentini

Displaying similar documents to “Semiflows and semigroups”

On the theory of remediability

Hassan Emamirad (2003)

Banach Center Publications

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Suppose ${G ₁ (t)}_{t \geq 0}$ and ${G ₂ (t)}_{t \geq 0}$ are two families of semigroups on a Banach space X (not necessarily of class C₀) such that for some initial datum u₀, G₁(t)u₀ tends towards an undesirable state u*. After remedying by means of an operator ρ we continue the evolution of the state by applying G₂(t) and after time 2t we retrieve a prosperous state u given by u = G₂(t)ρG₁(t)u₀. Here we are concerned with various properties of the semigroup (t): ρ → G₂(t)ρG₁(t). We define (X) to be the space of remedial operators...

Is $A^{- 1}$ an infinitesimal generator?

Hans Zwart (2007)

Banach Center Publications

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In this paper we study the question whether $A^{- 1}$ is the infinitesimal generator of a bounded C₀-semigroup if A generates a bounded C₀-semigroup. If the semigroup generated by A is analytic and sectorially bounded, then the same holds for the semigroup generated by $A^{- 1}$ . However, we construct a contraction semigroup with growth bound minus infinity for which $A^{- 1}$ does not generate a bounded semigroup. Using this example we construct an infinitesimal generator of a bounded semigroup for which its...

Linear modulus of a multidimensional semigroup of $L_{1}$ -contractions and a differentiation theorem

Dogan Çömez (1989)

Colloquium Mathematicae

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On a vector-valued local ergodic theorem in $L_{\infty}$

Ryotaro Sato (1999)

Studia Mathematica

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Let $T = T (u) : u \in ℝ_{d}^{+}$ be a strongly continuous d-dimensional semigroup of linear contractions on $L_{1} ((Ω, Σ, μ); X)$ , where (Ω,Σ,μ) is a σ-finite measure space and X is a reflexive Banach space. Since $L_{1} ((Ω, Σ, μ); X) * = L_{\infty} ((Ω, Σ, μ); X *)$ , the adjoint semigroup $T * = T * (u) : u \in ℝ_{d}^{+}$ becomes a weak*-continuous semigroup of linear contractions acting on $L_{\infty} ((Ω, Σ, μ); X *)$ . In this paper the local ergodic theorem is studied for the adjoint semigroup T*. Assuming that each T(u), $u \in ℝ_{d}^{+}$ , has a contraction majorant P(u) defined on $L_{1} ((Ω, Σ, μ); ℝ)$ , that is, P(u) is a positive linear contraction on $L_{1} ((Ω, Σ, μ); ℝ)$ such that $‖ T (u) f (ω) ‖ \leq P (u) ‖ f (\cdot) ‖ (ω)$ almost...

A general differentiation theorem for multiparameter additive processes

Ryotaro Sato (2002)

Colloquium Mathematicae

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Let $(L, | | \cdot | |_{L})$ be a Banach lattice of equivalence classes of real-valued measurable functions on a σ-finite measure space and $T = T (u) : u = (u ₁, . . ., u_{d}), u_{i} > 0, 1 \leq i \leq d$ be a strongly continuous locally bounded d-dimensional semigroup of positive linear operators on L. Under suitable conditions on the Banach lattice L we prove a general differentiation theorem for locally bounded d-dimensional processes in L which are additive with respect to the semigroup T.

Good and very good magnifiers

Marin Gutan (2000)

Bollettino dell'Unione Matematica Italiana

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Un elemento $a$ di un semigruppo $S$ è un elemento accrescitivo sinistro se la traslazione $λ_{a}$ di $S$ , associata all'elemento $a$ , è surgettiva e non è iniettiva (E. S. Ljapin, [13], § 5). Così, per ogni elemento accrescitivo sinistro $a$ , esiste un sottoinsieme proprio $M$ di $S$ tale che la restrizione a $M$ di $λ_{a}$ è biunivoca. Se $M$ è un sottosemigruppo (risp. un ideale destro) di $S$ , l'elemento accrescitivo sinistro $a$ viene detto buono (risp. molto buono) (F. Migliorini [15], [16], [17]). Utilizzando...

On ergodicity for operators with bounded resolvent in Banach spaces

Kirsti Mattila (2011)

Studia Mathematica

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We prove results on ergodicity, i.e. on the property that the space is a direct sum of the kernel of an operator and the closure of its range, for closed linear operators A such that ${| | α (α - A)}^{- 1} | |$ is uniformly bounded for all α > 0. We consider operators on Banach spaces which have the property that the space is complemented in its second dual space by a projection P. Results on ergodicity are obtained under a norm condition ||I - 2P|| ||I - Q|| < 2 where Q is a projection depending on the...

On semigroups with an infinitesimal operator

Jolanta Olko (2005)

Annales Polonici Mathematici

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Let $F^{t} : t \geq 0$ be an iteration semigroup of linear continuous set-valued functions. If the semigroup has an infinitesimal operator then it is a uniformly continuous semigroup majorized by an exponential semigroup. Moreover, for sufficiently small t every linear selection of $F^{t}$ is invertible and there exists an exponential semigroup $f^{t} : t \geq 0$ of linear continuous selections $f^{t}$ of $F^{t}$ .

Normal cryptogroups with an associate subgroup

Mario Petrich (2013)

Czechoslovak Mathematical Journal

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Let $S$ be a semigroup. For $a, x \in S$ such that $a = a x a$ , we say that $x$ is an associate of $a$ . A subgroup $G$ of $S$ which contains exactly one associate of each element of $S$ is called an associate subgroup of $S$ . It induces a unary operation in an obvious way, and we speak of a unary semigroup satisfying three simple axioms. A normal cryptogroup $S$ is a completely regular semigroup whose $ℋ$ -relation is a congruence and $S / ℋ$ is a normal band. Using the representation of $S$ as a strong semilattice of Rees matrix...

A $C^{*}$ -algebraic Schoenberg theorem

Ola Bratteli, Palle E. T. Jorgensen, Akitaka Kishimoto, Donald W. Robinson (1984)

Annales de l'institut Fourier

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Let $𝔄$ be a $C^{*}$ -algebra, $G$ a compact abelian group, $τ$ an action of $G$ by $*$ -automorphisms of $𝔄, 𝔄^{τ}$ the fixed point algebra of $τ$ and $𝔄_{F}$ the dense sub-algebra of $G$ -finite elements in $𝔄$ . Further let $H$ be a linear operator from $𝔄_{F}$ into $𝔄$ which commutes with $τ$ and vanishes on $𝔄^{τ}$ . We prove that $H$ is a complete dissipation if and only if $H$ is closable and its closure generates a $C_{0}$ -semigroup of completely positive contractions. These complete dissipations are classified in terms of certain twisted negative...

On the K-theory of the $C^{*}$ -algebra generated by the left regular representation of an Ore semigroup

Joachim Cuntz, Siegfried Echterhoff, Xin Li (2015)

Journal of the European Mathematical Society

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We compute the $K$ -theory of $C^{*}$ -algebras generated by the left regular representation of left Ore semigroups satisfying certain regularity conditions. Our result describes the $K$ -theory of these semigroup $C^{*}$ -algebras in terms of the $K$ -theory for the reduced group $C^{*}$ -algebras of certain groups which are typically easier to handle. Then we apply our result to specific semigroups from algebraic number theory.

A local Landau type inequality for semigroup orbits

Gerd Herzog, Peer Christian Kunstmann (2014)

Studia Mathematica

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Given a strongly continuous semigroup ${(S (t))}_{t \geq 0}$ on a Banach space X with generator A and an element f ∈ D(A²) satisfying $| | S (t) f | | \leq e^{- ω t} | | f | |$ and $| | S (t) A ² f | | \leq e^{- ω t} | | A ² f | |$ for all t ≥ 0 and some ω > 0, we derive a Landau type inequality for ||Af|| in terms of ||f|| and ||A²f||. This inequality improves on the usual Landau inequality that holds in the case ω = 0.

Strong mixing Markov semigroups on C₁ are meager

Wojciech Bartoszek, Beata Kuna (2006)

Colloquium Mathematicae

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We show that the set of those Markov semigroups on the Schatten class ₁ such that in the strong operator topology $l i m_{t \to \infty} T (t) = Q$ , where Q is a one-dimensional projection, form a meager subset of all Markov semigroups.

Inverses of generators of nonanalytic semigroups

Ralph deLaubenfels (2009)

Studia Mathematica

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Suppose A is an injective linear operator on a Banach space that generates a uniformly bounded strongly continuous semigroup ${e^{t A}}_{t \geq 0}$ . It is shown that $A^{- 1}$ generates an $O (1 + τ) A {(1 - A)}^{- 1}$ -regularized semigroup. Several equivalences for $A^{- 1}$ generating a strongly continuous semigroup are given. These are used to generate sufficient conditions on the growth of ${e^{t A}}_{t \geq 0}$ , on subspaces, for $A^{- 1}$ generating a strongly continuous semigroup, and to show that the inverse of -d/dx on the closure of its image in L¹([0,∞)) does not generate...