Motivated by some recent twaddles on Mazur rotations problem, we study the "dynamics" of the semigroup of contractive automorphisms of Banach spaces, mostly in finite-dimensional spaces. We focus on the metric aspects of the "action" of such semigroups, the size of the orbits and semitransitivity properties, and their impact on the geometry of the unit ball of the underlying space.
A result by Liskevich and Perelmuter from 1995 yields the optimal angle of analyticity for symmetric submarkovian semigroups on $L_p$, $1<p<\infty$. C.~Kriegler showed in 2011 that the result remains true without the assumption of positivity of the semigroup. Here we give an elementary proof of Kriegler's result.
We analyse $C_0$-semigroups of contractive operators on real-valued $L^p$-spaces for $p \not= 2$ and on other classes of non-Hilbert spaces. We show that, under some regularity assumptions on the semigroup, the geometry of the unit ball of those spaces forces the semigroup's generator to have only trivial (point) spectrum on the imaginary axis. This has interesting consequences for the asymptotic behaviour as $t \to \infty$. For example, we can show that a contractive and eventually norm continuous $C_0$-semigroup on a real-valued $L^p$-space automatically converges strongly if $p \not\in \{1,…
The space of unitary $C_{0}$-semigroups on separable infinite dimensional Hilbert space, when viewed under the topology of uniform weak convergence on compact subsets of $\mathbb{R}_{+}$, is known to admit various interesting residual subspaces. Before treating the contractive case, the problem of the complete metrisability of this space was raised in [Eisner, 2010]. Utilising Borel complexity computations and automatic continuity results for semigroups, we obtain a general result, which in particular implies that the one-/multiparameter contractive $C_{0}$-semigroups constitute Polish spaces …
Based on the technique of enriching contractive type mappings, a technique that has been used successfully in some recent papers, we introduce the concept of {\it saturated} class of contractive mappings. We show that, from this perspective, the contractive type mappings in the metric fixed point theory can be separated into two distinct classes, unsaturated and saturated, and that, for any unsaturated class of mappings, the technique of enriching contractive type mappings provides genuine new fixed point results. We illustrate the concept by surveying some significant fixed point results obta…
Working over infinite dimensional separable Hilbert spaces, residual results have been achieved for the space of contractive $C_{0}$-semigroups under the topology of uniform weak operator convergence on compact subsets of $\mathbb{R}_{+}$. Eisner and Serény raised in 2009 the open problem: Does this space constitute a Baire space? Observing that the subspace of unitary semigroups is completely metrisable and appealing to known density results, we solve this problem positively by showing that certain topological properties can in general be transferred from dense subspaces to larger spaces. The…
Choose an arbitrary but fixed set of $n\times n$ matrices $A_1, \ldots, A_m$ and let $Ω_\mathbf A\subset \mathbb C^m$ be the unit ball with respect to the norm $\|\cdot\|_{\mathbf A},$ where $\|(z_1,\ldots ,z_m)\|_{\mathbf A}=\|z_1A_1+ \cdots+z_mA_m\|_{\rm op}.$ It is known that if $m\geq 3$ and $\mathbb B$ is any ball in $\mathbb C^m$ with respect to some norm, say $\|\cdot\|_{\mathbb B},$ then there exists a contractive linear map $L:(\mathbb C^m,\|\cdot\|^*_{\mathbb B})\to \mathcal M_k$ which is not completely contractive. The characterization of those balls in $\mathbb C^2$ for which contr…
In this paper, we continue our previous research studies of exponential ultradistribution semigroups in Banach spaces. The existence and uniqueness of analytical solutions of abstract fractional relaxation equations associated with the generators of exponential ultradistribution semigroups have been considered. Some other results are also proved.
In this paper, we prove the existence of a common fixed point in $C(K)$, the Chebyshev center of K, for a left reversible semigroup of isometry mappings. This existence result improves the results obtained by Lim et al. and Brodskii and Milman.
Our aim in this paper is to present a new type of the modular space. This space contains the classical modular space. There are some mappings that do not have contractive condition in the usual modular space but become contraction in this new space.
We generalize results concerning $\mathrm{C}_0$-semigroups on Banach lattices to a setting of ordered Banach spaces. We prove that the generator of a disjointness preserving $\mathrm{C}_0$-semigroup is local. Some basic properties of local operators are also given. We investigate cases where local operators generate local $\mathrm{C}_0$-semigroups, by using Taylor series or Yosida approximations. As norms we consider regular norms and show that bands are closed with respect to such norms. Our proofs rely on the theory of embedding pre-Riesz spaces in vector lattices and on corresponding extens…
In this paper we obtain new characterizations of the uniformly convex and smooth Banach spaces. These characterizations are established by using Lp-boundedness properties of Littlewood-Paley functions and area integrals associated with heat semigroups and involving fractional derivatives. Our results apply for instance by considering the heat semigroups defined by Hermite and Laguerre operators that do not satisfy the Markovian property.
The first part of this article is a brief survey of the properties of so-called almost interior points in ordered Banach spaces. Those vectors can be seen as a generalization of ``functions which are strictly positive almost everywhere'' on $L^p$-spaces and of ``quasi-interior points'' in Banach lattices. In the second part we study the long--term behaviour of strongly positive operator semigroups on ordered Banach spaces; these are semigroups which, in a sense, map every non-zero positive vector to an almost interior point. Using the Jacobs--de Leeuw--Glicksberg decomposition together with th…
Let $A$ be the generator of a $C_0$-semigroup $T$ on a Banach space of analytic functions on the open unit disc. If $T$ consists of composition operators, then there exists a holomorphic function $G:{\mathbb D}\to{\mathbb C}$ such that $Af=Gf'$ with maximal domain. The aim of the paper is the study of the reciprocal implication.
The purpose of this paper is to revisit previous works of the author with J. Sjöstrand (2010--2021) proved in the Hilbert case by considering the Banach case at the light of a paper by Y.~Latushkin and V.~Yurov (2013).
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 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 $\mathbb R^n$, $n\geq 2$.
Let X be a real or complex Banach space and let F in X be a non-empty set. F is called an existence set of best coapproximation (existence set for brevity), if for any x in X, $R_F(x)$ is not the empty set, where $$ R_F (x) = \{ d \in F : \|d-c\| \leq \|x-c\| \hbox{ for any } c \in F \}.$$ It is clear that any existence set is a contractive subset of X. The aim of this paper is to present some conditions on F and X under which the notions of exsistence set and contractive set are equivalent.
Let $G$ be a locally compact group. We prove that the augmentation ideal in $L^1(G)$ is (algebraically) finitely-generated as a left ideal if and only if $G$ is finite. We then investigate weighted versions of this result, as well as a version for semigroup algebras. Weighted measure algebras are also considered. We are motivated by a recent conjecture of Dales and Żelazko, which states that a unital Banach algebra in which every maximal left ideal is finitely-generated is necessarily finite-dimensional. We prove that this conjecture holds for many of the algebras considered. Finally, we use t…
We study, to certain Banach spaces $X$, families of weighted composition operators. Notably, we show that if this family form a strongly continuous semigroup, then its infinitesimal generator ($Γ, D(Γ)$) is given by $Γf = gf+Gf^\prime$ with $D(Γ) = \{ f\in X \ | \ gf+Gf^\prime\in X \}$ where $g, G$ are holomorphic functions. Moreover, our second maim result is to study the reciprocal implication. That is if $(Γ, D(Γ))$, define like above, generate a strongly continuous semigroup, then this one is a family of weighted composition operators.
We give a relation between the exponential stability of $ C_{0}- $semigroup $ \textbf{T}=\left\lbrace T(t) \right\rbrace_{t\geq 0} $ and the solutions of Lyapunov inequality \( \left\langle QAx,x\right\langle +\left\langle Qx,Ax\right\langle \leq -||x||^{2}, \) in $ B^{+}(X,X^{*}) $, with $ X $ is a Banach space. The solutions of this inequality characterizes, the boundedness of the resolvent $ R(λ,A) $ inside and outside of the left half-plane $ \Reλ\geq 0 $, and also the left invertibility of the $ C_{0}- $semigroup T.
We characterize strong continuity of general operator semigroups on some Lebesgue spaces. In particular, a characterization of strong continuity of weighted composition semigroups on classical Hardy spaces and weighted Bergman spaces with regular weights is given. As applications, our result improves the results of Siskakis, A. G. \cite{AG1} and König, W. \cite{K} and answers a question of Siskakis, A. G. proposed in \cite{AG4}. We also characterize strongly continuous semigroups of weighted composition operators on weighted Bergman spaces in terms of abelian intertwiners of multiplication ope…
In this paper, we analyze recurrent $C_{0}$-semigroups of bounded operators on Banach spaces. We also introduce the notion of a (uniformly) $C_{0}$-rigid semigroups of bounded operators and give a structural characterization of them. A characterization of a Li-Yorke chaoticity of the translation semigroup $(T(t))_{t\geq 0}$ on weighted spaces of integrable functions and continuous functions in terms of admissible weight function is given. The recurrent $C_0$-semigroups induced by semiflows are characterized on the spaces of integrable functions and of spaces of continuous functions.
We develop a theory of eventually positive $C_0$-semigroups on Banach lattices, that is, of semigroups for which, for every positive initial value, the solution of the corresponding Cauchy problem becomes positive for large times. We give characterisations of such semigroups by means of spectral and resolvent properties of the corresponding generators, complementing existing results on spaces of continuous functions. This enables us to treat a range of new examples including the square of the Laplacian with Dirichlet boundary conditions, the bi-Laplacian on $L^p$-spaces, the Dirichlet-to-Neuma…
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…
We introduce a large class of contractive mappings, called Suzuki Berinde type contraction. We show that any Suzuki Berinde type contraction has a fixed point and characterizes the completeness of the underlying normed space. A fixed point theorem for multivalued mapping is also obtained. These results unify, generalize and complement various known comparable results in the literature.