Countable groups of isometries on Banach spaces

Uniqueness of complex structure and real hereditarily indecomposable Banach spaces

Let $X$ be a real reflexive, smooth and separable Banach space having the Kadeč-Klee property and compactly imbedded in the real Banach space $Y$ and let $G:Y\rightarrow \mathbb{R} $ be a differentiable functional. By using the "fountain theorem" (Bartsch [3]), we will study the multiplicity of solutions for the operator equation \[ J_{\varphi}u=G^{\prime}(u)\text{,} \] where $J_{\varphi}$ is the duality mapping on $X$, corresponding to the gauge function $\varphi$. Equations having the above form with $J_{\varphi}$ a duality mapping on Orlicz-Sobolev spaces are considered as applications. As particular cases of the latter results, some multiplicity results concerning duality mappings on Sobolev spaces are derived.

Let $X$ be a real reflexive and separable Banach space having the Kadeč-Klee property, compactly imbedded in the real Banach space $V$ and let $G\colon V\rightarrow {\mathbb R} $ be a differentiable functional. By using ``fountain theorem'' and ``dual fountain theorem'' (Bartsch [< i> Infinitely many solutions of a symmetric Dirichlet problem< /i> , Nonlinear Anal. < b> 20< /b> (1993), 1205–1216] and Bartsch-Willem [< i> On an elliptic equation with concave and convex nonlinearities< /i> , Proc. Amer. Math. Soc. < b> 123< /b> (1995), 3555–3561], respectively), we will study the multiplicity of solutions for operator equation $$ J_{\varphi}u=G^{\prime}(u), $$ where $J_{\varphi}$ is the duality mapping on $X$, corresponding to the gauge function $\varphi$. Equations having the above form with $J_{\varphi}$ a duality mapping on Orlicz-Sobolev spaces are considered as applications. As particular cases of the latter results, some multiplicity results concerning duality mappings on Sobolev spaces are derived.

Let B be a real separable Banach space and R: i'->ia covariance operator. All representations of R in the form 2en ® e, (e, n > 1} c fi, are characterized. Necessary and sufficient conditions for R to be compact are ob- tained, including a generalization of Mercer's theorem. An application to character- istic functions is given. 1. Introduction. The study of covariance operators is a major component in the theory of probability measures on Banach spaces (10), (9), (1). The covariance operator of a strong second-order measure is always compact (2); however, the covariance operator of a weak second-order measure need not be compact. In this paper we first characterize series representations of covariance operators, and then give a set of necessary and sufficient conditions for a covariance operator to be compact. The classical Mercer's theorem (7) can be obtained as an immediate corollary. These results are then applied to extend a result of Prohorov and Sazanov (6) on relative compactness of probability measures from Hubert space to Banach space. 2. Definitions and notation. B is a real separable Banach space with norm || ■ || and topological dual B*. A linear operator R: B* -» B is a covariance operator if 7? is symmetric and nonnegative: {Ru, u> = and 0, for all u, v in B*. A probability measure ii on the Borel a-field of B is said to be weak second-order if fB(x, «>2 dii(x) > = j = I {x — m, u}(x — m, v} d(i(x),

In this correspondence, complete convergence theorems are established for weighted row sums from arrays of random elements taking values in real separable Rademacher type Banach spaces as well as real separable martingale type Banach spaces. It is assumed that A version of the Rademacher type p complete convergence theorem is also established with random variable weights. Illustrative examples are included.

Let X be a separable real Banach space having a k-times con- tinuously Fr echet dierentiable (i.e. C k -smooth) norm where k 2 f1;:::;1g. We show that any equivalent norm on X can be approximated uniformly on bounded sets by C k -smooth norms.

Embedding of level-continuous fuzzy sets on Banach spaces

exists for each y ~ X. This condition is equivalent to the condition that a unique support functional f(x) = llfll = 1 (f ~ X*) exists at x. By the Mazur theorem, the points of smoothness of the sphere of each separable Banach space form a dense G6-subset of S (see [i]). Moreover, each separable Banach space admits an equivalent norm, in which each point of the new unit sphere is a point of smoothness [2, p. 25]. A point x ~ S will be called a fixed point of smoothness of S if it is a point of smoothness of the unit sphere SX** of the second adjoint space X** under the canonical embedding of X in X**. In addition, the norm is also said to be very smooth at the point x ~ S (see [3] and [2, p. 30]). It is clear that if a space is reflexive, then each point of smoothness of its unit sphere is fixed. If the norm is Frech~t-differentiable at a point x ~ S (i.e., the limit (i) exists uniformly with respect to all y ~ X), then x is a fixed point of smoothness (see [3] and [2, p. 30]). Each Banach space with separable adjoint admits an equivalent Frechet-differentiable norm [4], i.e., each point of the new unit sphere is a fixed point of smoothness in this norm. It may be asked as to howmany fixed points of smoothness convex bounded centrally symmetric bodies (unit balls in equivalent norms) of a separable Banach space have? A verbatim repetition of Mazur's arguments (see [i]) in relation to the space X** shows that if the space X has separable second adjoint, then the fixed points of smoothness of its unit sphere form a dense G6-subset of S. However the following theorem is valid.

Fix an even integer d≥2. Let X be a separable real Banach space. Here we prove the existence a closed subspace Y of X and homogeneous degree d polynomials Q, Q 1, respectively on X and on Y with the following properties: i. Q(x) ≥ 0 for all x∈X and Q(x)=0 if and only if x∈Y; ii. Q 1(y)>0 for all y∈Y; iii. Q 1 is strictly convex, i.e. for every ε > 0 the set B(Q 1,ε):= {y∈Y: Q 1(y)≤ε} is convex and Q 1(tu+(1-t)v))<ε for all u, v∈B(Q 1,ε) and all 0 <t <1; iv. the continuous homogeneous form Q˜ induced by Q on X / Y is strictly convex. In the complex case we prove a similer existence result for complete intersections of the projects spaces P(Y) and P(X/Y).

Geometric duality theory of cones in dual pairs of vector spaces

In this paper we study when the Daugavet equation is satisfied for weakly compact polynomials on a Banach space X, i.e. when the equality ‖Id + P‖ = 1 + ‖P‖ is satisfied for all weakly compact polynomials P : X −→ X. We show that this is the case when X = C(K), the real or complex space of continuous functions on a compact space K without isolated points. We also study the alternative Daugavet equation max |ω|=1 ‖Id + ω P‖ = 1 + ‖P‖ for polynomials P : X −→ X. We show that this equation holds for every polynomial on the complex space X = C(K) (K arbitrary) with values in X. The result is not true in the real case. Finally, we study the Daugavet and the alternative Daugavet equations for k-homogeneous polynomials. In 1963, I. K. Daugavet [13] showed that every compact linear operator T on C[0, 1] satisfies ‖Id + T‖ = 1 + ‖T‖, a norm equality which has currently become known as the Daugavet equation. Over the years, the validity of the above equality has been established for many classes of operators on many Banach spaces. For instance, weakly compact linear operators on C(K), K perfect, and L1(μ), μ atomless, satisfy Daugavet equation (see [25] for an elementary approach). We refer the reader to the books [1, 2] and the papers [20, 26] for more information and background. It is also a remarkable result given in 1970 by J. Duncan et al. [16] that, for every compact Hausdorff space K and every bounded linear operator T on C(K), the equality max ω∈T ‖Id + ωT‖ = 1 + ‖T‖ holds, where we use T to denote the unit sphere of the base field. The above norm equality is now known as the alternative Daugavet equation [22], and it is satisfied by all bounded linear operators on C(K) and L1(μ), K and μ arbitrary. We refer the reader to [16, 21, 22] and references there in for background. The aim of this paper is to study the Daugavet equation and the alternative Daugavet equation for polynomials in Banach spaces. There is a concept, the numerical range of an operator (see below for the definition), intimately related to the Daugavet and alternative Daugavet equations. The definition of numerical range for bounded linear operators on Banach spaces was given in 1962 by F. Bauer [5] (see [9, 10] for background) extending the 1918 classical definition of numerical range (or field of values) of a matrix given by O. Toeplitz [24]. In 1968, the concept of numerical range was extended to arbitrary continuous functions from the unit sphere of a real or complex Banach space into the space by F. Bonsall, B. Cain, and H. Schneider [7]. In the seventies, Date: March 24th, 2006; revised September 13th, 2006. 2000 Mathematics Subject Classification. Primary 46G25; Secondary 46B20, 47A12.

A closed, convex and bounded setP in a Banach spaceE is called a polytope if every finite-dimensional section ofP is a polytope. A Banach spaceE is called polyhedral ifE has an equivalent norm such that its unit ball is a polytope. We prove here: (1) LetW be an arbitrary closed, convex and bounded body in a separable polyhedral Banach spaceE and let e>0. Then there exists a tangential e-approximating polytopeP for the bodyW. (2) LetP be a polytope in a separable Banach spaceE. Then, for every e>0,P can be e-approximated by an analytic, closed, convex and bounded bodyV.

Analytic Approximations of Uniformly Continuous Functions in Real Banach Spaces

Strassen's version of the law of the iterated logarithm is proved for Brownian motion in a real separable Banach space. We apply this result to obtain the law of the iterated logarithm for a sequence of independent Gaussian random variables with values in a Banach space and to obtain Strassen's result. Introduction. Let H denote a real separable Hilbert space with norm 11*IH and assume llIIB is a measurable norm on H in the sense of (2). Then there exists a constant M > 0 such that IIXIIB 0, let m, denote the canonical Gaussian cylinder set measure on H with variance parameter t and let t1t (t > 0) denote the Borel probability measure on B induced by m, (t > 0). We call , the Wiener measure on B generated by H with variance parameter t. Let SIB denote the space of continuous functions w from (0, om) into B such that w(0) = 0, and let 9 be the a-field of 2B generated by the functions w -* w(t).

Cylindrical Wiener processes in real separable Banach spaces are defined, and an approximation theorem involving scalar Wiener processes is given for such processes. A weak stochastic integral for Banach spaces involving a cylindrical Wiener process as integrator and an operator-valued stochastic process as integrand is defined. Basic properties of this integral are stated and proved. A class of linear, time-invariant, stochastic differential equations in real, separable, reflexive Banach spaces is formulated in such fashion that a solution of the equation is a cylindrical process. An existence and uniqueness theorem is proved. A stochastic version of the problem of heat conduction in a ring provides an example.