The structure of uniformly Gâteaux smooth Banach spaces

It is shown that a Banach spaceX admits an equivalent uniformly Gateaux smooth norm if and only if the dual ball ofX* in its weak star topology is a uniform Eberlein compact. Supported by AV 101-97-02, AV 1019003 and GA ČR 201-98-1449. Supported by GA ČR 201-98-1449, AV 1019003 and GAUK 1/1998.     

On the range of the derivative of Gâteaux-smooth functions on separable banach spaces

We prove that there exists a Lipschitz function froml ¹ into ℝ² which is Gateaux-differentiable at every point and such that for everyx, y εl ¹, the norm off′(x) −f′(y) is bigger than 1. On the other hand, for every Lipschitz and Gateaux-differentiable function from an arbitrary Banach spaceX into ℝ and for everyε > 0, there always exist two pointsx, y εX such that ‖f′(x) −f′(y)‖ is less thanε. We also construct, in every infinite dimensional separable Banach space, a real valued functionf onX, which is Gateaux-differentiable at every point, has bounded non-empty support, and with the properties thatf′ is norm to weak* continuous andf′(X) has an isolated pointa, and that necessarilya ε 0. This work has been initiated while the second-named author was visiting the University of Bordeaux. The second-named author is supported by grant AV 1019003, A1 019 205, GA CR 201 01 1198.     

Gâteaux differentiability and uniform monotone approximation of convex functions in Banach spaces

Acta Mathematica Hungarica - We prove that if $$X^{*}$$ is strictly convex, a convex function $$f$$ is coercive and b-Lipschitzian iff there exist two convex function sequences...     

Darboux property of Gâteaux derivatives of functions on ℝ n

D. Preiss proved that the graph of the derivative of a continuous Gateaux-differentiable function f : ℝ² → ℝ is always connected. We show that this is no longer true in higher dimensions: we construct a continuous, Gateaux-differentiable function f : ℝ³ → ℝ for which the range of its gradient mapping {∇ f(x) : x ∈ ℝ³} is disconnected. We also give an example of an approximately differentiable continuous function on ℝ² such that the range of its gradient mapping is disconnected. The work is a part of the research project MSM 0021620839 financed by MSMT and it was also partly supported by GAČR 201/06/0198 and GAČR 201/06/0018.     

On (N,ϵ)-pseudospectra of operators on Banach spaces

In this paper we extend the concept of the $(N,?)$-pseudospectra of Hansen to the case of bounded linear operators on Banach spaces and prove several relations to the usual spectrum. We particularly discuss the approximation by rectangular finite sections and the impact of the fundamental result of Shargorodsky on “jumping” pseudospectra.     

On K_0-groups of operator algebras on Banach spaces

This paper merges some classifications of G-M-type Banach spaces simplifically, discusses the condition of K0(B(X)) = 0 for operator algebra B(X) on a Banach space X, and obtains a result to improve Laustsen's sufficient condition, gives an example to show that X ≈ X2 is not a sufficient condition of K0(B(X)) = 0.     

The isometrical extensions of 1-Lipschitz mappings on Gteaux differentiability spaces

Let X and Y be real Banach spaces.Suppose that the subset sm[S1(X)] of the smooth points of the unit sphere [S1(X)] is dense in S1(X).If T0 is a surjective 1-Lipschitz mapping between two unit spheres,then,under some condition,T0 can be extended to a linear isometry on the whole space.     

On property (β) in Banach lattices, Calderón-Lozanowskii and Orlicz-Lorentz spaces

The geometry of Calderón-Lozanowskii spaces, which are strongly connected with the interpolation theory, was essentially developing during the last few years (see [4, 9, 10, 12, 13, 17]). On the other hand many authors investigated property (β) in Banach spaces (see [7, 19, 20, 21, 25, 26]). The first aim of this paper is to study property (β) in Banach function lattices. Namely a criterion for property (β) in Banach function lattice is presented. In particular we get that in Banach function lattice property (β) implies uniform monotonicity. Moreover, property (β) in generalized Calderón-Lozanowskii function spaces is studied. Finally, it is shown that in Orlicz-Lorentz function spaces property (β) and uniform convexity coincide.     

On S-asymptotically ω-periodic functions on Banach spaces and applications

This paper is devoted to the study of the class of continuous and bounded functions for which exists ω>0 such that limt→∞(f(t+ω)−f(t))=0 (in the sequel called S-asymptotically ω-periodic functions). We discuss qualitative properties and establish some relationships between this type of functions and the class of asymptotically ω-periodic functions. We also study the existence of S-asymptotically ω-periodic mild solutions of the first-order abstract Cauchy problem in Banach spaces.     

S(n)-θ-closed spaces

A P-space is called P-θ-closed if it is θ-closed in every P-space in which it is embedded. S(n)-θ-closed spaces are characterized in terms of special covers and filters, and a categorical property of S(n)-θ-closed spaces is established. Examples are given to distinguish S(n)-θ-closed and S(n)-closed spaces and to show that the property of being S(n)-θ-closed is not productive.     

Itô formula for stochastic integrals w.r.t. compensated Poisson random measures on separable Banach spaces

We prove the Ito formula (1.3) for Banach valued functions acting on stochastic processes with jumps, the martingale part given by stochastic integrals of time dependent Banach valued random functions w.r.t. compensated Poisson random measures. Such stochastic integrals have been discussed by Mandrekar and Rüdiger, Stochastics and Stochastic Reports 78(4), 189–212 (2006) and Rüdiger (2004), Stochastics and Stochastic Reports, 76, pp. 213–242.     

On left (right) ρ-transitive geometric spaces

In this article, first we generalize the concept of $\mathfrak{B}$-parts in a geometric space with respect to a binary relation $\rho$ and then we study left (right) $l{\mathfrak{B}}_{\rho }$-closures ($r{\mathfrak{B}}_{\rho }$-closures, respectively). Finally we introduce and investigate the notions left $\rho$-strongly transitive geometric spaces and left $\rho$-emphatic strongly transitive geometric spaces.     

The generalized Roper-Suffridge extension operator in Banach spaces (Ⅲ)

Suppose that X is a complex Banach space with the norm ||·|| and n is a positive integer with dim X ≥n≥ 2. In this paper,we consider the generalized Roper-Suffridge extension operator Φn,β2,γ2,...,βn+1,γn+1(f) on the domain Ωp1,p2,...,pn+1 defined by     

Existence and uniqueness of path wise solutions for stochastic integral equations driven by Lévy noise on separable Banach spaces

The stochastic integrals of M- type 2 Banach valued random functions w.r.t. compensated Poisson random measures introduced in (Rüdiger, B., 2004, In: Stoch. Stoch. Rep., 76, 213–242.) are discussed for general random functions. These are used to solve stochastic integral equations driven by non Gaussian Lévy noise on such spaces. Existence and uniqueness of the path wise solutions are proven under local Lipshitz conditions for the drift and noise coefficients on M-type 2 as well as general separable Banach spaces. The continuous dependence of the solution on the initial data as well as on the drift and noise coefficients are shown. The Markov properties for the solutions are analyzed.     

Dual spaces of C∞ (X)

ForX a locally compact Stonian Space, letC _∞ (X) denote the universally complete Riesz space of all extended-real-valued continuous functionsf onX for which {x∈X| |f (x)|=∞} is nowhere dense. In this paper the dual spaces ofC _∞ (X) (i.e. the spaces of order bounded; of σ-order continuous; of order continuous linear forms onC _∞ (X), and the extended order dual ofC _∞ (X) denote here byC _∞ (X)^ρ (introduced by W.A.J. Luxemburg and J.J. Masterson)) are characterized. It is shown thatC _∞ (X)^ρ can be identified in a canonical way with the inductive limitM _q (X) of the Riesz spaces of all normal Radon measures defined on the dense open subsets ofX. More generally, ifY is a locally compact space thenM _q (Y) is the extended order dual of the inductive limit of the Riesz spaces of all real-valued continuous functions defined on the dense open subsets ofY. IfX is locally compact and hyperstonian, then it is proved thatC _∞ (X) andC _∞ (X)^ρ are isomorphic, and a criterion forC _∞ (X)^ρ to be the universal completion of the space of order continuous linear forms onC _∞ (X) is given.     

On dynamics of Out(F n ) on PSL2(ℂ) characters

We introduce and study an open set of PSL₂(?) characters of a nonabelian free group, on which the action of the outer automorphism group is properly discontinuous, and which is strictly larger than the set of discrete, faithful convex-cocompact (i.e. Schottky) characters. This implies, in particular, that the outer automorphism group does not act ergodically on the set of characters with dense image. Hence there is a difference between the geometric (discrete vs. dense) decomposition of the characters, and a natural dynamical decomposition.     

On the Congruence σ(n)≡1(mod n)

Let k≥2 be an integer,and let σ(n) denote the sum of the positive divisors of an integer n.We call n a quasi-multiperfect number if σ(n)=kn+1.In this paper,we give some necessary properties of them.