Linear modulus of a multidimensional semigroup of $L_1$-contractions and a differentiation theorem

Dogan Çömez

Displaying similar documents to “Linear modulus of a multidimensional semigroup of $L_{1}$ -contractions and a differentiation theorem”

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...

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...

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...

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 $F^{t}$ .

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.

The algebra of the subspace semigroup of $M ₂ (_{q})$

Jan Okniński (2002)

Colloquium Mathematicae

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The semigroup $S = S (M ₂ (_{q}))$ of subspaces of the algebra $M ₂ (_{q})$ of 2 × 2 matrices over a finite field $_{q}$ is studied. The ideal structure of S, the regular -classes of S and the structure of the complex semigroup algebra ℂ[S] are described.

Locally adequate semigroup algebras

Yingdan Ji, Yanfeng Luo (2016)

Open Mathematics

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We build up a multiplicative basis for a locally adequate concordant semigroup algebra by constructing Rukolaĭne idempotents. This allows us to decompose the locally adequate concordant semigroup algebra into a direct product of primitive abundant [...] 0-J* $0 - 𝒥 *$ -simple semigroup algebras. We also deduce a direct sum decomposition of this semigroup algebra in terms of the [...] ℛ* $ℛ *$ -classes of the semigroup obtained from the above multiplicative basis. Finally, for some special cases, we...

Spaces of multipliers and their preduals for the order multiplication on [0, 1]

Savita Bhatnagar, H. L. Vasudeva (2002)

Colloquium Mathematicae

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Let I = [0, 1] be the compact topological semigroup with max multiplication and usual topology. C(I), $L^{p} (I)$ , 1 ≤ p ≤ ∞, are the associated Banach algebras. The aim of the paper is to characterise $H o m_{C (I)} (L^{r} (I), L^{p} (I))$ and their preduals.

Some model theory of SL(2,ℝ)

Jakub Gismatullin, Davide Penazzi, Anand Pillay (2015)

Fundamenta Mathematicae

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We study the action of G = SL(2,ℝ), viewed as a group definable in the structure M = (ℝ,+,×), on its type space $S_{G} (M)$ . We identify a minimal closed G-flow I and an idempotent r ∈ I (with respect to the Ellis semigroup structure * on $S_{G} (M)$ ). We also show that the “Ellis group” (r*I,*) is nontrivial, in fact it is the group with two elements, yielding a negative answer to a question of Newelski.

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.

On upper triangular nonnegative matrices

Yizhi Chen, Xian Zhong Zhao, Zhongzhu Liu (2015)

Czechoslovak Mathematical Journal

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We first investigate factorizations of elements of the semigroup $S$ of upper triangular matrices with nonnegative entries and nonzero determinant, provide a formula for $ρ (S)$ , and, given $A \in S$ , also provide formulas for $l (A)$ , $L (A)$ and $ρ (A)$ . As a consequence, open problem 2 and problem 4 presented in N. Baeth et al. (2011), are partly answered. Secondly, we study the semigroup of upper triangular matrices with only positive integral entries, compute some invariants of such semigroup, and also partly answer...

Fractional powers of operators, K-functionals, Ulyanov inequalities

Walter Trebels, Ursula Westphal (2010)

Banach Center Publications

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Given an equibounded (₀)-semigroup of linear operators with generator A on a Banach space X, a functional calculus, due to L. Schwartz, is briefly sketched to explain fractional powers of A. Then the (modified) K-functional with respect to $(X, D ({(- A)}^{α}))$ , α > 0, is characterized via the associated resolvent R(λ;A). Under the assumption that the resolvent satisfies a Nikolskii type inequality, ${| | λ R (λ; A) f | |}_{Y} {\leq c φ (1 / λ) | | f | |}_{X}$ , for a suitable Banach space Y, an Ulyanov inequality is derived. This will be of interest if one has...

On some free semigroups, generated by matrices

Piotr Słanina (2015)

Czechoslovak Mathematical Journal

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Let $A = [\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}], B_{λ} = [\begin{matrix} 1 & 0 \\ λ & 1 \end{matrix}] .$ We call a complex number $λ$ “semigroup free“ if the semigroup generated by $A$ and $B_{λ}$ is free and “free” if the group generated by $A$ and $B_{λ}$ is free. First families of semigroup free $λ$ ’s were described by J. L. Brenner, A. Charnow (1978). In this paper we enlarge the set of known semigroup free $λ$ ’s. To do it, we use a new version of “Ping-Pong Lemma” for semigroups embeddable in groups. At the end we present most of the known results related to semigroup free and free numbers in a common...

Semi-Characters of the Multiplicative Semigroup of Integers Modulo $m$

Bohumír Parízek, Štefan Schwarz (1961)

Matematicko-fyzikálny časopis

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The one-sided ergodic Hilbert transform in Banach spaces

Guy Cohen, Christophe Cuny, Michael Lin (2010)

Studia Mathematica

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Let T be a power-bounded operator on a (real or complex) Banach space. We study the convergence of the one-sided ergodic Hilbert transform $l i m_{n} \sum_{k = 1}^{n} (T^{k} x) / k$ . We prove that weak and strong convergence are equivalent, and in a reflexive space also $s u p_{n} | | \sum_{k = 1}^{n} (T^{k} x) / k | | < \infty$ is equivalent to the convergence. We also show that $- \sum_{k = 1}^{\infty} (T^{k}) / k$ (which converges on (I-T)X) is precisely the infinitesimal generator of the semigroup ${(I - T)}_{| \bar{(I - T) X}}^{r}$ .

On the probability that two elements of a finite semigroup have the same right matrix

Attila Nagy, Csaba Tóth (2022)

Commentationes Mathematicae Universitatis Carolinae

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We study the probability that two elements which are selected at random with replacement from a finite semigroup $S$ have the same right matrix.

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.

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...

Displaying similar documents to “Linear modulus of a multidimensional semigroup of L 1 -contractions and a differentiation theorem”

Displaying similar documents to “Linear modulus of a multidimensional semigroup of $L_{1}$ -contractions and a differentiation theorem”