In this study, we refine the compactification presented by Witz \cite{Witz} for general semigroups to the case of bounded $C_0$-semigroups, involving adjoint theory for this class of operators. This approach considerably reduces the operator space in which the compactification is performed. Additionally, this approach leads to a decomposition of $X^{\sun}$ and to an extension of ergodic results to dual semigroups.
The given study uses the methods to identify compactifications of semigroups $S\subset L(X),$ which reside in the space $L(X).$ This method generalizes in some sense the deLeeuw-Glicksberg-Theory to a greater class of functions. The approach provides an abstract approach to several notions of almost periodicity, which mainly involving right semitopological semigroups \cite{RuppertLNM}, and the adjoint theory. Moreover, the given setting is refined to the case of bounded $C_0-$semigroups.
We introduce the notion of an introverted Boolean algebra $\cal B$ of closed-and-open subsets of a topological group $G$, show that the associated Stone space $(ν_{\cal B} G, ν_{\cal B})$ is a totally disconnected semigroup compactification of $G$, and show that every totally disconnected semigroup compactification of $G$ takes this form. We identify and study the universal totally disconnected semigroup compactification, the universal totally disconnected semitopological semigroup compactification and the universal totally disconnected group compactification of $G$. Our main results are obtai…
Our aim is to find some new links between linear (circular) orderability of groups and topological dynamics. We suggest natural analogs of the concept of algebraic orderability for topological groups involving order-preserving actions on compact spaces and the corresponding enveloping semigroups in the sense of R. Ellis. This approach leads to several natural questions. Some of them might be useful also for discrete (countable) orderable groups.
In this short note, we give some new results on continuous bounded cohomology groups of topological semigroups with values in complex field. We show that the second continuous bounded cohomology group of a compact metrizable semigroup, is a Banach space. Also, we study cohomology groups of amenable topological semigroups, and we show that cohomology groups of rank greater than one of a compact left or right amenable semigroup, are trivial. Also, we give some examples and applications about topological lattices.
In this paper, we first study some properties peculiar to $C_{0}$--semigroups of continuous module homomorphisms and give a characterization for such a $C_{0}$--semigroup to be almost surely bounded. Then, based on these, we establish the Hille-Yosida generation theorem for almost surely bounded $C_{0}$--semigroups of continuous module homomorphisms, which generalizes some known results. Moreover, the counterexample constructed in this paper also shows that it is necessary to require the almost sure boundedness for such $C_{0}$--semigroups.
We discuss the notion of $γ$-$H^{\infty}$-bounded calculus, strong $γ$-$m$-$H^{\infty}$-bounded calculus on half-plane and weak-$γ$-Gomilko-Shi-Feng condition and give a connection between them. Then we state a characterization of generation of $γ$-bounded $C_0$-semigroup in $K$-convex space, which leads to a version of Gearhart-Prüss on $K$-convex space.
Let $0<α\leq 1$. We prove that a $α$-Kreiss bounded $C_0$ semigroup $(T_t)_{t \ge 0}$ on a Hilbert space has asymptotics $||T_t|| = O\big(t^α/\sqrt{log(t)}\big)$. Then, we give an application to perturbed wave equation.
In this paper, we prove that the generator of any bounded analytic semigroup in $(θ,1)$-type real interpolation of its domain and underlying Banach space has maximal $L^1$-regularity, using a duality argument combined with the result of maximal continuous regularity. As an application, we consider maximal $L^1$-regularity of the Dirichlet-Laplacian and the Stokes operator in inhomogeneous $B^s_{q,1}$-type Besov spaces on domains of $\mathbb R^n$, $n\geq 2$.
The paper deals with multidimensional Bochner-Phillips functional calculus. In the previous paper by the author bounded perturbations of Bernstein functions of several commuting semigroup generators on Banach spaces where considered, conditions for Lipschitzness and estimates for the norm of commutators of such functions where proved. Also in the one-dimensional case the Frechet differentiability of Bernstein functions of semigroup generators on Banach spaces where proved and a generalization of Livschits-Kreĭn trace formula derived. The aim of the present paper is to prove the Frechet differe…
In this paper, we consider positive Desch-Schappacher perturbations of bi-continuous semigroups on $\mathrm{AM}$-spaces with an additional property concerning the additional locally convex topology. As an examples, we discuss perturbations of the left-translation semigroup on the space of bounded continuous function on the real line and on the space of bounded linear operators.
We construct a new bounded functional calculus for the generators of bounded semigroups on Hilbert spaces and generators of bounded holomorphic semigroups on Banach spaces. The calculus is a natural (and strict) extension of the classical Hille-Phillips functional calculus, and it is compatible with the other well-known functional calculi. It satisfies the standard properties of functional calculi, provides a unified and direct approach to a number of norm-estimates in the literature, and allows improvements of some of them.
We prove a Desch-Schappacher type perturbation theorem for one-parameter semigroups on Banach spaces which are not strongly continuous for the norm, but possess a weaker continuity property. In this paper we chose to work in the framework of bi-continuous semigroups. This choice has the advantage that we can treat in a unified manner two important classes of semigroups: implemented semigroups on the Banach algebra $\mathscr{L}(E)$ of bounded, linear operators on a Banach space $E$, and semigroups on the space of bounded and continuous functions over a Polish space induced by jointly continuous…
This article summarises the theory of several bounded functional calculi for unbounded operators that have recently been discovered. The extend the Hille--Phillips calculus for (negative) generators $A$ of certain bounded $C_0$-semigroups, in particular for bounded semigroups on Hilbert spaces and bounded holomorphic semigroups on Banach spaces. They include functions outside the Hille-Phillips class, and they generally give sharper bounds for the norms of the resulting operators $f(A)$. The calculi are mostly based on appropriate reproducing formulas for the relevant classes of functions, and…
A small part of real line which is very close to zero has rich combinatorial properties. The aim of this paper is to express and then prove some locally combinatorial concepts near a virtual idempotent by considering the $wap$-compactification of a semitopological semigroup $S$. The $wap-$compactification of a semitopological semigroup $S$, is denoted by $S^w$, the collection of all ultrafilters near an idempotent $η\in S^w$ forms a compact subsemigroup of $βS_d$, where $S_d$ denotes $S$ as discrete space.
In this paper, we derive the $L^p$-boundedness of the variation operators associated with the heat semigroup which is generated by the high order Schrödinger type operator $(-Δ)^2+V^2$. Further more, we prove the boundedness of the variation operators on Morrey spaces. In the proof of the main results, we always make use of the variation inequalities associated with the heat semigroup generated by the biharmonic operator $(-Δ)^2.$
In this paper, we compare several Cesàro and Kreiss type boundedness conditions for a $C_0$-semigroup on a Banach space and we show that those conditions are all equivalent for a positive semigroup on a Banach lattice. Furthermore, we give an estimate of the growth rate of a Kreiss bounded and eventually positive $C_0$-semigroup $(T_t)_{t\ge 0}$ on certain Banach lattices $X$. We prove that if $X$ is an $L^p$-space, $1<p<+\infty$, then $\|T_t\| = \mathcal{O}\left(t/\log(t)^{\max(1/p,1/p')}\right)$ and if $X$ is an $(\text{AL})$ or $(\text{AM})$-space, then $\|T_t\|=\mathcal{O}(t^{1-ε})$ for so…
We investigate off-diagonal decay properties of the generalized Stokes semigroup with bounded measurable coefficients on $\mathrm{L}^2_σ (\mathbb{R}^d)$. Such estimates are well-known for elliptic equations in the form of pointwise heat kernel bounds and for elliptic systems in the form of integrated off-diagonal estimates. On our way to unveil this off-diagonal behavior we prove resolvent estimates in Morrey spaces $\mathrm{L}^{2 , ν} (\mathbb{R}^d)$ with $0 \leq ν< 2$.
Consider $d$ commuting $C_{0}$-semigroups (or equivalently: $d$-parameter $C_{0}$-semigroups) over a Hilbert space for $d \in \mathbb{N}$. In the literature (\textit{cf.} [29, 26, 27, 23, 18, 25]), conditions are provided to classify the existence of unitary and regular unitary dilations. Some of these conditions require inspecting values of the semigroups, some provide only sufficient conditions, and others involve verifying sophisticated properties of the generators. By focussing on semigroups with bounded generators, we establish a simple and natural condition on the generators, \textit{viz…
In this paper we prove weighted $\ell^p$-inequalities for variation and oscillation operators defined by semigroups of operators associated with discrete Jacobi operators. Also, we establish that certain maximal operators involving sums of differences of discrete Jacobi semigroups are bounded on weighted $\ell^p$-spaces. $\ell^p$-boundedness properties for the considered operators provide information about the convergence of the semigroup of operators defining them.
The growth-fragmentation equation models systems of particles that grow and reproduce as time passes. An important question concerns the asymptotic behaviour of its solutions. Bertoin and Watson ($2018$) developed a probabilistic approach relying on the Feynman-Kac formula, that enabled them to answer to this question for sublinear growth rates. This assumption on the growth ensures that microscopic particles remain microscopic. In this work, we go further in the analysis, assuming bounded fragmentations and allowing arbitrarily small particles to reach macroscopic mass in finite time. We esta…
In this paper we analyze the convergence of the following type of series \begin{equation*} T_N f(x)=\sum_{j=N_1}^{N_2} v_j\Big(\mathcal{P}_{a_{j+1}} f(x)-\mathcal{P}_{a_{j}} f(x)\Big),\quad x\in \mathbb R_+, \end{equation*} where $\{\mathcal{P}_t \}_{t>0}$ is the Poisson semigroup of the Bessel operator $\displaystyle Δ_λ:=-{d^2\over dx^2}-{2λ\over x}{d\over dx}$ with $λ$ being a positive constant, $N=(N_1, N_2)\in \mathbb Z^2$ with $N_1<N_2,$ $\{v_j\}_{j\in \mathbb Z}$ is a bounded real sequences and $\{a_j\}_{j\in \mathbb Z}$ is an increasing real sequence. {Our analysis will consist in the …
In this note we connect the notion of solutions of a martingale problem to the notion of a strongly continuous and locally equi-continuous semigroup on the space of bounded continuous functions equipped with the strict topology. This extends the classical connection of semigroups to Markov processes that was used successfully in the context of compact spaces to the context of Polish spaces. In addition, we consider the context of locally compact spaces and show how the transition semigroup on the space of functions vanishing at infinity can be extended to the space of bounded continuous functi…
In this paper we establish $L^p$-boundedness properties for variation, oscillation and jump operators associated with Riesz transforms and Poisson semigroups related to Laguerre polynomial expansions.
We prove that for every semigroup of Schwarz maps on the von~Neumann algebra of all bounded linear operators on a Hilbert space which has a subinvariant faithful normal state there exists an associated semigroup of contractions on the space of Hilbert-Schmidt operators of the Hilbert space. Moreover, we show that if the original semigroup is weak$^*$ continuous then the associated semigroup is strongly continuous. We introduce the notion of the extended generator of a semigroup on the bounded operators of a Hilbert space with respect to an orthonormal basis of the Hilbert space. We describe th…
In this paper we study the behavior of some harmonic analysis operators associated with the discrete Laplacian $Δ_d$ in discrete Hardy spaces $\mathcal H^p(\mathbb Z)$. We prove that the maximal operator and the Littlewood-Paley $g$ function defined by the semigroup generated by $Δ_d$ are bounded from $\mathcal H^p(\mathbb Z)$ into $\ell^p(\mathbb Z)$, $0<p\leq 1$. Also, we establish that every $Δ_d$-spectral multiplier of Laplace transform type is a bounded operator from $\mathcal H^p(\mathbb Z)$ into itself, for every $0<p\leq 1$.
