Suppose A is an injective linear operator on a Banach space that generates a uniformly bounded strongly continuous semigroup ${e^{tA}}_{t\ge 0}$. It is shown that $A^{-1}$ generates an $O(1+\tau )A{(1-A)}^{-1}$-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 ${e^{tA}}_{t\ge 0}$, 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...

Given a strongly continuous semigroup ${\left(S\left(t\right)\right)}_{t\ge 0}$ on a Banach space X with generator A and an element f ∈ D(A²) satisfying $\left|\right|S\left(t\right)f\left|\right|\le e^{-\omega t}\left|\right|f\left|\right|$ and $\left|\right|S\left(t\right)A²f\left|\right|\le e^{-\omega t}\left|\right|A²f\left|\right|$ 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.

We study the notion of left $\varphi$-biflatness for Segal algebras and semigroup algebras. We show that the Segal algebra $S\left(G\right)$ is left $\varphi$-biflat if and only if $G$ is amenable. Also we characterize left $\varphi$-biflatness of semigroup algebra $l^1\left(S\right)$ in terms of biflatness, when $S$ is a Clifford semigroup.

We give sufficient conditions on an operator space E and on a semigroup of operators on a von Neumann algebra M to obtain a bounded analytic or R-analytic semigroup (${\left(T\otimes I{d}_E\right)}_{t\ge 0}$ on the vector valued noncommutative $L^p$-space $L^p(M,E)$. Moreover, we give applications to the $H^{\infty }\left({\Sigma }_{\theta }\right)$ functional calculus of the generators of these semigroups, generalizing some earlier work of M. Junge, C. Le Merdy and Q. Xu.

We consider discrete dynamical systems whose phase spaces are compact metrizable countable spaces. In the first part of the article, we study some properties that guarantee the continuity of all functions of the corresponding Ellis semigroup. For instance, if every accumulation point of $X$ is fixed, we give a necessary and sufficient condition on a point $a\in X^{\text{'}}$ in order that all functions of the Ellis semigroup $E(X,f)$ be continuous at the given point $a$. In the second part, we consider transitive...

Let be a locally compact Hausdorff space. Let $A_i$, i = 0,1,...,N, be generators of Feller semigroups in C₀() with related Feller processes $X_i=X_i\left(t\right),t\ge 0$ and let ${\alpha }_i$, i = 0,...,N, be non-negative continuous functions on with ${\sum }_{i=0}^N{\alpha }_i=1$. Assume that the closure A of ${\sum }_{k=0}^N{\alpha }_k{A}_k$ defined on ${\bigcap }_{i=0}^N\left(A_i\right)$ generates a Feller semigroup T(t), t ≥ 0 in C₀(). A natural interpretation of a related Feller process X = X(t), t ≥ 0 is that it evolves according to the following heuristic rules: conditional on being at a point p ∈ , with probability...

Let $S$ be a numerical semigroup. We say that $h\in \mathbb{N}\setminus S$ is an isolated gap of $S$ if $\{h-1,h+1\}\subseteq S.$ A numerical semigroup without isolated gaps is called a perfect numerical semigroup. Denote by $\mathrm{m}\left(S\right)$ the multiplicity of a numerical semigroup $S$. A covariety is a nonempty family $𝒞$ of numerical semigroups that fulfills the following conditions: there exists the minimum of $𝒞,$ the intersection of two elements of $𝒞$ is again an element of $𝒞$, and $S\setminus \left\{\mathrm{m}\right(S\left)\right\}\in 𝒞$ for all $S\in 𝒞$ such that $S\ne min\left(𝒞\right).$ We prove that the set $𝒫\left(F\right)=\{S:S$ is a perfect numerical semigroup...

We characterise the boundedness of the $H^{\infty }$ calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if $-A$ generates a bounded analytic $C_0$ semigroup $\left(T_t\right)$ on a UMD space, then the $H^{\infty }$ calculus of $A$ is bounded if and only if $\left(T_t\right)$ has a dilation to a bounded group on $L^2([0,1],X)$. This generalises a Hilbert space result of C.LeMerdy. If $X$ is an $L^p$ space we can choose another $L^p$ space in place of $L^2([0,1],X)$.

Let $X$ be a Banach space of analytic functions on the open unit disk and $\Gamma$ a subset of linear isometries on $X$. Sufficient conditions are given for non-supercyclicity of $\Gamma$. In particular, we show that the semigroup of linear isometries on the spaces $S^p$ ($p>1$), the little Bloch space, and the group of surjective linear isometries on the big Bloch space are not supercyclic. Also, we observe that the groups of all surjective linear isometries on the Hardy space $H^p$ or the Bergman space $L_a^p$ ($1<p<\infty$, $p\ne 2$)...

Let Ω be an open subset of ${ℝ}^d$ with 0 ∈ Ω. Furthermore, let $H_{\Omega }=-{\sum }_{i,j=1}^d{\partial }_i{c}_{ij}{\partial }_j$ be a second-order partial differential operator with domain $C_c^{\infty }\left(\Omega \right)$ where the coefficients $c_{ij}\in W_{loc}^{1,\infty }\left(\Omega ̅\right)$ are real, $c_{ij}=c_{ji}$ and the coefficient matrix $C=\left(c_{ij}\right)$ satisfies bounds 0 < C(x) ≤ c(|x|)I for all x ∈ Ω. If ${\int }_0^{\infty }ds{s}^{d/2}{e}^{-\lambda \mu \left(s\right)²}<\infty$ for some λ > 0 where $\mu \left(s\right)={\int }_0^{s}dtc{\left(t\right)}^{-1/2}$ then we establish that $H_{\Omega }$ is L₁-unique, i.e. it has a unique L₁-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e. it has a unique L₂-extension which generates a submarkovian semigroup....

We show that the growth rates of solutions of the abstract differential equations ẋ(t) = Ax(t), $ẋ\left(t\right)=A^{-1}x\left(t\right)$, and the difference equation $x_d(n+1)=(A+I){(A-I)}^{-1}{x}_d\left(n\right)$ are closely related. Assuming that A generates an exponentially stable semigroup, we show that on a general Banach space the lowest growth rate of the semigroup ${\left(e^{A^{-1}t}\right)}_{t\ge 0}$ is O(∜t), and for $\left((A+I){(A-I)}^{-1}\right)ⁿ$ it is O(∜n). The similarity in growth holds for all Banach spaces. In particular, for Hilbert spaces the best estimates are O(log(t)) and O(log(n)), respectively. Furthermore,...

Let $\left[n\right]=\{1,...,n\}$ be an $n$-chain. We give presentations for the following transformation semigroups: the semigroup of full order-decreasing mappings of $\left[n\right]$, the semigroup of partial one-to-one order-decreasing mappings of $\left[n\right]$, the semigroup of full order-preserving and order-decreasing mappings of $\left[n\right]$, the semigroup of partial one-to-one order-preserving and order-decreasing mappings of $\left[n\right]$, and the semigroup of partial order-preserving and order-decreasing mappings of $\left[n\right]$.

We give new results on square functions $${\parallel x\parallel }_F=\parallel {\left({\int }_0^{\infty }{\left|F\left(tA\right)x\right|}^2\frac{\mathrm{d}t}{t}\right)}^{1/2}{\parallel }_p$$ associated to a sectorial operator $A$ on $L^p$ for $1\&lt;p\&lt;\infty$. Under the assumption that $A$ is actually $R$-sectorial, we prove equivalences of the form $K^{-1}{\parallel x\parallel }_G\le {\parallel x\parallel }_F\le K{\parallel x\parallel }_G$ for suitable functions $F,G$. We also show that $A$ has a bounded $H^{\infty }$ functional calculus with respect to ${\parallel .\parallel }_F$. Then we apply our results to the study of conditions under which we have an estimate $\parallel {({\int }_0^{\infty }|C{\mathrm{e}}^{-tA}{\left(x\right)|}^2\mathrm{d}t{)}^{1/2}\parallel }_q\le M{\parallel x\parallel }_p$, when $-A$ generates a bounded semigroup ${\mathrm{e}}^{-tA}$ on $L^p$ and $C:D\left(A\right)\to L^q$ is a linear mapping.

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.

Completely regular semigroups $\mathrm{𝒞ℛ}$ are considered here with the unary operation of inversion within the maximal subgroups of the semigroup. This makes $\mathrm{𝒞ℛ}$ a variety; its lattice of subvarieties is denoted by $ℒ\left(\mathrm{𝒞ℛ}\right)$. We study here the relations $𝐊,T,L$ and $𝐂$ relative to a sublattice $\Psi$ of $ℒ\left(\mathrm{𝒞ℛ}\right)$ constructed in a previous publication. For $𝐑$ being any of these relations, we determine the $𝐑$-classes of all varieties in the lattice $\Psi$ as well as the restrictions of $𝐑$ to $\Psi$.

Let $X$ be a Banach space, $ℬ\left(X\right)$ the algebra of bounded linear operators on $X$ and $(J,\parallel ·{\parallel }_J)$ an admissible Banach ideal of $ℬ\left(X\right)$. For $T\in ℬ\left(X\right)$, let $L_{J,T}$ and $R_{J,T}\in ℬ\left(J\right)$ denote the left and right multiplication defined by $L_{J,T}\left(A\right)=TA$ and $R_{J,T}\left(A\right)=AT$, respectively. In this paper, we study the transmission of some concepts related to recurrent operators between $T\in ℬ\left(X\right)$, and their elementary operators $L_{J,T}$ and $R_{J,T}$. In particular, we give necessary and sufficient conditions for $L_{J,T}$ and $R_{J,T}$ to be sequentially recurrent. Furthermore, we prove that $L_{J,T}$ is recurrent...