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We give a survey on the different regularity properties of sectorial operators on Banach spaces. We present the main results and open questions in the theory and then concentrate on the known methods to construct various counterexamples.
Let $X$ be a compact Hausdorff space, let $E$ be a Banach space, and let $C(X,E)$ stand for the Banach space of $E$-valued continuous functions on $X$ under the uniform norm. In this paper we characterize Integral operators (in the sense of…
In this paper, we introduce the concept of isotone cones in Banach spaces. Then we apply the order monotonic property of the metric projection operator to prove the existence of best approximations for some operators without continuity…
We investigate some types of composition operators, linear and not, and conditions for some spaces to be mapped into themselves and for the operators to satisfy some good properties.
An algebra A of operators on a Banach space X is called strictly semi-transitive if for all non-zero x,y in X there exists an operator S in A such that Sx=y or Sy=x. We show that if A is norm-closed and strictly semi-transitive, then every…
We show that some previous results concerning the boundedness of differentiation and integration operators on weighted spaces given by radial weights in the unit disk or the complex plane might fail without some natural additional…
In this article we extend recent results by the first author on the necessity of $BMO$ for the boundedness of commutators on the classical Lebesgue spaces. We generalize these results to a large class of Banach function spaces. We show that…
Here I review recent work, by other authors and by myself, on some particular topics related to the regular and chaotic motion in elliptical galaxies. I show that it is quite possible to build highly stable triaxial stellar systems that…
We produce several situations where some natural subspaces of classical Banach spaces of functions over a compact abelian group contain the space $c_0$.
We describe the proper closed invariant subspaces of the integration operator when it acts continuously on countable intersections and countable unions of weighted Banach spaces of holomorphic functions on the unit disc or the complex…
A detailed theory of stochastic integration in UMD Banach spaces has been developed recently by the authors. The present paper is aimed at giving various sufficient conditions for stochastic integrability.
The purpose of the present work is to treat a new notion related to linear dynamics, which can be viewed as a "localization" of the notion of hypercyclicity. In particular, let $T$ be a bounded linear operator acting on a Banach space $X$…
In this paper, we study the cyclicity of the shift operator $S$ acting on a Banach space $\X$ of analytic functions on the open unit disc $\D$. We develop a general framework where a method based on a corona theorem can be used to show that…
A 3D-dynamical model is constructed for the study of motion in the central regions of a disk galaxy with a double nucleus. Using the results of the 2D-model, we find the regions of initial conditions in the (x,px,z,py)=EJ, (y=pz=0) phase…
Recent work on the dynamics of triaxial stellar systems is reviewed. The motion of boxlike orbits in realistic triaxial potentials is generically stochastic. The degree to which the stochasticity manifests itself in the dynamics depends on…
Various properties of the (continuous) Ces\`aro operator $\mathsf{C}$, acting on Banach and Fr\'echet spaces of continuous functions and $L^p$-spaces, are investigated. For instance, the spectrum and point spectrum of $\mathsf{C}$ are…
We consider group-valued cocycles over dynamical systems with hyperbolic behavior. The base system is either a hyperbolic diffeomorphism or a mixing subshift of finite type. The cocycle $A$ takes values in the group of invertible bounded…
We provide sufficient conditions for a Banach space Y to be weakly sequentially complete. These conditions are expressed in terms of the existence of directional derivatives for cone convex mappings with values in Y .
We study properties of the topological space of composition operators on the Banach algebra of bounded functions on an unbounded, locally finite metric space in the operator norm topology and essential norm topology. Moreover, we…
We compute the lattice operations for the (pairwise) stable set in many-to-many matching markets when only path-independence on agents' choice functions is imposed. To do this, we first show that the sets of firm-quasi-stable and…