Showing papers on "Banach space published in 2005"

Variational Analysis and Generalized Differentiation II

Abstract: Applications.- Constrained Optimization and Equilibria.- Optimal Control of Evolution Systems in Banach Spaces.- Optimal Control of Distributed Systems.- Applications to Economics.

The Mathematics of Arbitrage

Abstract: A Guided Tour to Arbitrage Theory.- The Story in a Nutshell.- Models of Financial Markets on Finite Probability Spaces.- Utility Maximisation on Finite Probability Spaces.- Bachelier and Black-Scholes.- The Kreps-Yan Theorem.- The Dalang-Morton-Willinger Theorem.- A Primer in Stochastic Integration.- Arbitrage Theory in Continuous Time: an Overview.- The Original Papers.- A General Version of the Fundamental Theorem of Asset Pricing (1994).- A Simple Counter-Example to Several Problems in the Theory of Asset Pricing (1998).- The No-Arbitrage Property under a Change of Numeraire (1995).- The Existence of Absolutely Continuous Local Martingale Measures (1995).- The Banach Space of Workable Contingent Claims in Arbitrage Theory (1997).- The Fundamental Theorem of Asset Pricingfor Unbounded Stochastic Processes (1998).- A Compactness Principle for Bounded Sequences of Martingales with Applications (1999).

Well-posed linear systems

Abstract: Many infinite-dimensional linear systems can be modelled in a Hilbert space setting. Others, such as those dealing with heat transfer or population dynamics, need to be set more generally in Banach spaces. This is the first book dealing with well-posed infinite-dimensional linear systems with an input, a state, and an output in a Hilbert or Banach space setting. It is also the first to describe the class of non-well-posed systems induced by system nodes. The author shows how standard finite-dimensional results from systems theory can be extended to these more general classes of systems, and complements them with new results which have no finite-dimensional counterpart. Much of the material presented is original, and many results have never appeared in book form before. A comprehensive bibliography rounds off this work which will be indispensable to all working in systems theory, operator theory, delay equations and partial differential equations.

Banach spaces adapted to Anosov systems

Abstract: We study the spectral properties of the Ruelle-Perron-Frobenius operator associated to an Anosov map on classes of functions with high smoothness. To this end we construct anisotropic Banach spaces of distributions on which the transfer operator has a small essential spectrum. In the C ∞ case, the essential spectral radius is arbitrarily small, which yieldsa descriptionof the correlationswith arbitraryprecision. Moreover,we obtain sharp spectral stability results for deterministic and random perturbations. In particular, we obtain differentiability results for spectral data (which imply differentiability of the Sinai-Ruelle-Bowenmeasure, the variancefor the centrallimit theorem, the rates of decay for smooth observable, etc.).

The Generic chaining : upper and lower bounds of stochastic processes

Abstract: Overview and Basic Facts.- Gaussian Processes and Related Structures.- Matching Theorems.- The Bernoulli Conjecture.- Families of distances.- Applications to Banach Space Theory.

Strong convergence of modified Mann iterations

Abstract: The Mann iterations for nonexpansive mappings have only weak convergence even in a Hilbert space. We propose two modifications of the Mann iterations in a uniformly smooth Banach space, one for nonexpansive mappings and the other for the resolvent of accretive operators. The two modified Mann iterations are proved to have strong convergence.

Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces

Abstract: In 1979, Ishikawa proved a strong convergence theorem for finite families of nonexpansive mappings in general Banach spaces. Motivated by Ishikawa's result, we prove strong convergence theorems for infinite families of nonexpansive mappings.

Stochastic integration of functions with values in a Banach space

Abstract: Let H be a separable real Hilbert space and let E be a real Banach space. In this paper we construct a stochastic integral for certain operator-valued functions Φ : (0, T ) → L (H,E) with respect to a cylindrical Wiener process {WH(t)}t∈[0,T ]. The construction of the integral is given by a series expansion in terms of the stochastic integrals for certain E-valued functions. As a substitute for the Itô isometry we show that the square expectation of the integral equals the radonifying norm of an operator which is canonically associated with the integrand. We obtain characterizations for the class of stochastically integrable functions and prove various convergence theorems. The results are applied to the study of linear evolution equations with additive cylindrical noise in general Banach spaces. An example is presented of a linear evolution equation driven by a one-dimensional Brownian motion which has no weak solution. In this paper we construct a theory of stochastic integration of operator-valued functions with respect to a cylindrical Wiener process. The range space of the operators is allowed to be an arbitrary real Banach space E. A stochastic integral of this type can be used for solving the linear stochastic Cauchy problem (0.1) dU(t) = AU(t) dt+B dWH(t), t ∈ [0, T ], U(0) = u0. Here, A is the infinitesimal generator of a strongly continuous semigroup {S(t)}t>0 of bounded linear operators on E, the operator B is a bounded linear operator from a separable real Hilbert space H into E, and {WH(t)}t∈[0,T ] is a cylindrical H-Wiener process. Formally, equation (0.1) is solved by the stochastic convolution process (0.2) U(t) = S(t)u0 + ∫ t 0 S(t− s)B dWH (s). It is well known that the integral on the right hand side can be interpreted as an Itô stochastic integral if E is a Hilbert space. A comprehensive theory of abstract stochastic differential equations in Hilbert spaces is presented in the monograph by Da Prato and Zabczyk [5]. More generally the integral can be defined for spaces E with martingale type 2. This has been worked out by Brzeźniak [2]. Examples of martingale type 2 spaces are Hilbert spaces and the Lebesgue spaces L(μ) with p ∈ [2,∞). In both settings, the integral is defined for step functions first, and for general functions the integral is obtained by a limiting argument. Such a limiting argument depends on 2000 Mathematics Subject Classification. Primary: 60H05; Secondary: 28C20, 35R15, 47D06, 60H15.

A Fixed Point Approach to Stability of a Quadratic Equation

Abstract: Using the fixed point alternative theorem we establish the orthogonal stability of quadratic functional equation of Pexider type $f(x+y)+g(x-y)=h(x)+k(y)$, where $f, g, h, k$ are mappings from a symmetric orthogonality space to a Banach space, by orthogonal additive mappings under a necessary and sufficient condition on $f$.

Proximal normal structure and relatively nonexpansive mappings

Abstract: The notion of proximal normal structure is introduced and used to study mappings that are "relatively nonexpansive" in the sense that they are defined on the union of two subsets A and B of a Banach space X and satisfy k T x − T yk ≤ k x − yk for all x ∈ A, y ∈ B. It is shown that if A and B are weakly compact and convex, and if the pair (A, B) has proximal normal structure, then a relatively nonexpansive mapping T : A ∪ B → A ∪ B satisfying (i) T(A) ⊆ B and T(B) ⊆ A, has a proximal point in the sense that there exists x0 ∈ A ∪ B such that k x0 − T x0k = dist(A, B). If in addition the norm of X is strictly convex, and if (i) is replaced with (i) ' T(A) ⊆ A and T(B) ⊆ B, then the conclusion is that there exist x0 ∈ A and y0 ∈ B such that x0 and y0 are fixed points of T and k x0 − y0k = dist(A, B). Because every bounded closed convex pair in a uniformly convex Banach space has proximal normal structure, these results hold in all uniformly convex spaces. A Krasnosel'skiuo type iteration method for approximating the fixed points of relatively nonexpansive mappings is also given, and some related Hilbert space results are discussed.

Regularization of ill-posed problems in Banach spaces: convergence rates

Abstract: This paper deals with quantitative aspects of regularization for ill-posed linear equations in Banach spaces, when the regularization is done using a general convex penalty functional. The error estimates shown here by means of Bregman distances yield better convergences rates than those already known for maximum entropy regularization, as well as for total variation regularization.

Continuous Frames, Function Spaces, and the Discretization Problem

Abstract: A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous superpositions. Associated to a given continuous frame we construct certain Banach spaces. Many classical function spaces can be identified as such spaces. We provide a general method to derive Banach frames and atomic decompositions for these Banach spaces by sampling the continuous frame. This is done by generalizing the coorbit space theory developed by Feichtinger and Grochenig. As an important tool the concept of localization of frames is extended to continuous frames. As a byproduct we give a partial answer to the question raised by Ali, Antoine, and Gazeau whether any continuous frame admits a corresponding discrete realization generated by sampling.

On metric Ramsey-type phenomena

Abstract: The main question studied in this article may be viewed as a nonlinear analogue of Dvoretzky?s theorem in Banach space theory or as part of Ramsey theory in combinatorics. Given a finite metric space on n points, we seek its subspace of largest cardinality which can be embedded with a given distortion in Hilbert space. We provide nearly tight upper and lower bounds on the cardinality of this subspace in terms of n and the desired distortion. Our main theorem states that for any
> 0, every n point metric space contains a subset of size at least n1-
which is embeddable in Hilbert space with O
log(1/
)
 distortion. The bound on the distortion is tight up to the log(1/
) factor. We further include a comprehensive study of various other aspects of this problem.

Nonlinear Semigroups, Fixed Points, and Geometry of Domains in Banach Spaces

Abstract: # Mappings in Metric and Normed Spaces # Differentiable and Holomorphic Mappings in Banach Spaces # Hyperbolic Metrics on Domains in Complex Banach Spaces # Some Fixed Point Principles # The Denjoy-Wolff Fixed Point Theory # Generation Theory for One-Parameter Semigroups # Flow-Invariance Conditions # Stationary Points of Continuous Semigroups # Asymptotic Behavior of Continuous Flows # Geometry of Domains in Banach Spaces

Proximal Methods in Vector Optimization

Abstract: We consider the vector optimization problem of finding weakly efficient points for maps from a Hilbert space X to a Banach space Y with respect to the partial order induced by a closed, convex, and pointed cone $C\subset Y$ with a nonempty interior. We develop for this problem an extension of the proximal point method for scalar-valued convex optimization. In this extension, the subproblems consist of finding weakly efficient points for suitable regularizations of the original map. We present both an exact and an inexact version, in which the subproblems are solved only approximately, within a constant relative tolerance. In both cases, we prove weak convergence of the generated sequence to a weakly efficient point, assuming convexity of the map with respect to C and C-completeness of the initial section. In cases where this last assumption fails, we still establish that the generating sequence is, in a suitable sense, a minimizing one. We also exhibit a particular instance of the algorithm for which, under a mild hypothesis on C, the weak limit of the generated sequence is an efficient, rather than a weakly efficient, point.

On σ-porous sets in abstract spaces

Abstract: The main aim of this survey paper is to give basic information about properties and applications of σ-porous sets in Banach spaces (and some other infinite-dimensional spaces). This paper can be considered a partial continuation of the author's 1987 survey on porosity and σ-porosity and therefore only some results, remarks, and references (important for infinite-dimensional spaces) are repeated. However, this paper can be used without any knowledge of the previous survey. Some new results concerning σ-porosity in finite-dimensional spaces are also briefly mentioned. However, results concerning porosity (but not σ-porosity) are mentioned only exceptionally.

Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces

Abstract: The existence and structure of uniform attractors in $V$ is proved for nonautonomous 2D Navier-stokes equations on bounded domain with a new class of external forces, termed normal in $L_{l o c}^2(\mathbb R; H)$ (see Definition 3.1), which are translation bounded but not translation compact in $L_{l o c}^2(\mathbb R; H)$. To this end, some abstract results are established. First, a characterization on the existence of uniform attractor for a family of processes is presented by the concept of measure of noncompactness as well as a method to verify it. Then, the structure of the uniform attractor is obtained by constructing skew product flow on the extended phase space with weak topology. Finally, the uniform attractor of a process is identified with that of a family of processes with symbols in the closure of the translation family of the original symbol in a Banach space with weak topology.

Large Deviations Asymptotics and the Spectral Theory of Multiplicatively Regular Markov Processes

Abstract: In this paper we continue the investigation of the spectral theory and exponential asymptotics of primarily discrete-time Markov processes, following Kontoyiannis and Meyn (2003). We introduce a new family of nonlinear Lyapunov drift criteria, which characterize distinct subclasses of geometrically ergodic Markov processes in terms of simple inequalities for the nonlinear generator. We concentrate primarily on the class of multiplicatively regular Markov processes, which are characterized via simple conditions similar to (but weaker than) those of Donsker-Varadhan. For any such process $\{\Phi(t)\}$ with transition kernel $P$ on a general state space $X$, the following are obtained. Spectral Theory: For a large class of (possibly unbounded) functionals $F$ on $X$, the kernel $\hat P(x,dy) = e^{F(x)} P(x,dy)$ has a discrete spectrum in an appropriately defined Banach space. It follows that there exists a "maximal," well-behaved solution to the "multiplicative Poisson equation," defined as an eigenvalue problem for $\hat P$. Multiplicative Mean Ergodic Theorem: Consider the partial sums of this process with respect to any one of the functionals $F$ considered above. The normalized mean of their moment generating function (and not the logarithm of the mean) converges to the above maximal eigenfunction exponentially fast. Multiplicative regularity: The Lyapunov drift criterion under which our results are derived is equivalent to the existence of regeneration times with finite exponential moments for the above partial sums. Large Deviations: The sequence of empirical measures of the process satisfies a large deviations principle in a topology finer that the usual tau-topology, generated by the above class of functionals. The rate function of this LDP is the convex dual of logarithm of the above maximal eigenvalue, and it is shown to coincide with the Donsker-Varadhan rate function in terms of relative entropy. Exact Large Deviations Asymptotics: The above partial sums are shown to satisfy an exact large deviations expansion, analogous to that obtained by Bahadur and Ranga Rao for independent random variables.

Topics in Banach space integration

Abstract: * Bochner Integral * Dunford and Pettis Integrals * McShane and Henstock-Kurzweil Integrals * More on the McShane Integral * Comparison of the Bochner and McShane Integrals * Comparison of the Pettis and McShane Integrals * Primitive of the McShane and Henstock-Kurzweil Integrals * Generalizations of Some Integrals

Metric Cotype

Abstract: We introduce the notion of cotype of a metric space, and prove that for Banach spaces it coincides with the classical notion of Rademacher cotype This yields a concrete version of Ribe's theorem, settling a long standing open problem in the nonlinear theory of Banach spaces We apply our results to several problems in metric geometry Namely, we use metric cotype in the study of uniform and coarse embeddings, settling in particular the problem of classifying when L_p coarsely or uniformly embeds into L_q We also prove a nonlinear analog of the Maurey-Pisier theorem, and use it to answer a question posed by Arora, Lovasz, Newman, Rabani, Rabinovich and Vempala, and to obtain quantitative bounds in a metric Ramsey theorem due to Matousek

Weak and strong convergence of a scheme with errors for two nonexpansive mappings

Abstract: In this paper, we are concerned with the study of an iterative scheme with errors involving two nonexpansive mappings. We approximate the common fixed points of these two mappings by weak and strong convergence of the scheme in a uniformly convex Banach space under a condition weaker than compactness.

Intrinsic localization of frames

Abstract: Several concepts for the localization of a frame are studied. The intrinsic localization of a frame is defined by the decay properties of its Gramian matrix. Our main result asserts that the canonical dual frame possesses the same intrinsic localization as the original frame. The proof relies heavily on Banach algebra techniques, in particular on recent spectral invariance properties for certain Banach algebras of infinite matrices. Intrinsically localized frames extend in a natural way to Banach frames for a class of associated Banach spaces which are defined by weighted lp-coefficients of their frame expansions. As an example, the time--frequency concentration of distributions is characterized by means of localized (nonuniform) Gabor frames.

On Discretization Schemes for Stochastic Evolution Equations

Abstract: Stochastic evolutional equations with monotone operators are considered in Banach spaces. Explicit and implicit numerical schemes are presented. The convergence of the approximations to the solution of the equations is proved.

On bases in Banach spaces

Abstract: We investigate various kinds of bases in infinite dimensional Banach spaces. In particular, we consider the complexity of Hamel bases in separable and non-separable Banach spaces and show that in a separable Banach space a Hamel basis cannot be analytic, whereas there are non-separable Hilbert spaces which have a discrete and closed Hamel basis. Further we investigate the existence of certain complete minimal systems in `∞ as well as in separable Banach spaces. Outline. The paper is concerned with bases in infinite dimensional Banach spaces. The first section contains the definitions of the various kinds of bases and biorthogonal systems and also summarizes some set-theoretic terminology and notation which will be used throughout the paper. The second section provides a survey of known or elementary results. The third section deals with Hamel bases and contains some consistency results proved using the forcing technique. The fourth section is devoted to complete minimal systems (including Φ-bases and Auerbach bases) and the last section contains open problems. ∗The research for this paper began during the Workshop on Set Theory, Topology, and Banach Space Theory, which took place in June 2003 at Queen’s University Belfast, whose hospitality is gratefully acknowledged. The workshop was supported by the Nuffield Foundation Grant NAL/00513/G of the third author, the EPSRC Advanced Fellowship of the second author and the grant GACR 201/03/0933 of the fourth author. 1