Geometry Of Banach Spaces Research Articles

Delta‐points and their implications for the geometry of Banach spaces

AbstractWe show that the Lipschitz‐free space with the Radon–Nikodým property and a Daugavet point recently constructed by Veeorg is in fact a dual space isomorphic to . Furthermore, we answer an open problem from the literature by showing that there exists a superreflexive space, in the form of a renorming of , with a ‐point. Building on these two results, we are able to renorm every infinite‐dimensional Banach space to have a ‐point. Next, we establish powerful relations between existence of ‐points in Banach spaces and their duals. As an application, we obtain sharp results about the influence of ‐points for the asymptotic geometry of Banach spaces. In addition, we prove that if is a Banach space with a shrinking ‐unconditional basis with , or if is a Hahn–Banach smooth space with a dual satisfying the Kadets–Klee property, then and its dual fail to contain ‐points. In particular, we get that no Lipschitz‐free space with a Hahn–Banach smooth predual contains ‐points. Finally, we present a purely metric characterization of the molecules in Lipschitz‐free spaces that are ‐points, and we solve an open problem about representation of finitely supported ‐points in Lipschitz‐free spaces.

The finite Jung constant in Banach spaces

AbstractWe study in this paper the finite Jung constant, its interplay with Kottman’s constant and its meaning regarding the geometry of Banach spaces.

Trace Operator on von Koch’s Snowflake

AbstractWe study properties of the boundary trace operator on the Sobolev space $$W^1_1(\Omega )$$ W 1 1 ( Ω ) . Using the density result by Koskela and Zhang (Arch. Ration. Mech. Anal. 222(1), 1-14 2016), we define a surjective operator $$Tr: W^1_1(\Omega _K)\rightarrow X(\Omega _K)$$ T r : W 1 1 ( Ω K ) → X ( Ω K ) , where $$\Omega _K$$ Ω K is von Koch’s snowflake and $$X(\Omega _K)$$ X ( Ω K ) is a trace space with the quotient norm. Since $$\Omega _K$$ Ω K is a uniform domain whose boundary is Ahlfors-regular with an exponent strictly bigger than one, it was shown by L. Malý (2017) that there exists a right inverse to Tr, i.e. a linear operator $$S: X(\Omega _K) \rightarrow W^1_1(\Omega _K)$$ S : X ( Ω K ) → W 1 1 ( Ω K ) such that $$Tr \circ S= Id_{X(\Omega _K)}$$ T r ∘ S = I d X ( Ω K ) . In this paper we provide a different, purely combinatorial proof based on geometrical structure of von Koch’s snowflake. Moreover we identify the isomorphism class of the trace space as $$\ell _1$$ ℓ 1 . As an additional consequence of our approach we obtain a simple proof of the Peetre’s theorem (Special Issue 2, 277-282 1979) about non-existence of the right inverse for domain $$\Omega $$ Ω with regular boundary, which explains Banach space geometry cause for this phenomenon.

Stationary phase method, powers of functions, and applications to functional analysis

The utility of the (weighted) van der Corput inequalities or of the stationary phase method is illustrated with various examples borrowed from: differentiability issues (Riemann’s function and related); functional analysis on Banach spaces or algebras of analytic functions (composition operators); and local Banach space geometry (Schäffer’s problem).

The Rochberg garden

In 1996, it was published the seminal work of Rochberg “Higher order estimates in complex interpolation theory” (Rochberg, 1996). Among many other things, the paper contains a new method to construct new Banach spaces having an intriguing behaviour: they are simultaneously interpolation spaces and twisted sums of increasing complexity. The fundamental idea of Rochberg is to consider for each z∈S the space formed by the arrays of the truncated sequence of the Taylor coefficients of the elements of the Calderón space. What was probably unforeseen is that the Rochberg constructions would lead to a deep theory connecting Interpolation theory, Homology, Operator Theory and the Geometry of Banach spaces. This work aims to synthetically present such connections, an up-to-date account of the theory and a list of significative open problems.

Geometry of Banach Spaces: A New Route Towards Position Based Cryptography

In this work we initiate the study of position based quantum cryptography (PBQC) from the perspective of geometric functional analysis and its connections with quantum games. The main question we are interested in asks for the optimal amount of entanglement that a coalition of attackers have to share in order to compromise the security of any PBQC protocol. Known upper bounds for that quantity are exponential in the size of the quantum systems manipulated in the honest implementation of the protocol. However, known lower bounds are only linear. In order to deepen the understanding of this question, here we propose a position verification (PV) protocol and find lower bounds on the resources needed to break it. The main idea behind the proof of these bounds is the understanding of cheating strategies as vector valued assignments on the Boolean hypercube. Then, the bounds follow from the understanding of some geometric properties of particular Banach spaces, their type constants. Under some regularity assumptions on the former assignment, these bounds lead to exponential lower bounds on the quantum resources employed, clarifying the question in this restricted case. Known attacks indeed satisfy the assumption we make, although we do not know how universal this feature is. Furthermore, we show that the understanding of the type properties of some more involved Banach spaces would allow to drop out the assumptions and lead to unconditional lower bounds on the resources used to attack our protocol. Unfortunately, we were not able to estimate the relevant type constant. Despite that, we conjecture an upper bound for this quantity and show some evidence supporting it. A positive solution of the conjecture would lead to stronger security guarantees for the proposed PV protocol providing a better understanding of the question asked above.

Complexity of distances: Reductions of distances between metric and Banach spaces

We show that all the standard distances from metric geometry and functional analysis, such as Gromov-Hausdorff distance, Banach-Mazur distance, Kadets distance, Lipschitz distance, Net distance, and Hausdorff-Lipschitz distance have all the same complexity and are reducible to each other in a precisely defined way. This is done in terms of descriptive set theory and is a part of a larger research program initiated by the authors in \emph{Complexity of distances: Theory of generalized analytic equivalence relations}. The paper is however targeted also to specialists in metric geometry and geometry of Banach spaces.

Revisiting the minimum-norm problem

The design of optimal Magnetic Resonance Imaging (MRI) coils is modeled as a minimum-norm problem (MNP), that is, as an optimization problem of the form min_{xinmathcal{R}}|x|, where mathcal{R} is a closed and convex subset of a normed space X. This manuscript is aimed at revisiting MNPs from the perspective of Functional Analysis, Operator Theory, and Banach Space Geometry in order to provide an analytic solution to the following MRI problem: min_{psiinmathcal{R}}|psi|_{2}, where mathcal{R}:={psiin mathbb{R}^{n}:frac{|Apsi-b|_{infty}}{|b|_{infty}} leq D}, with Ainmathcal{M}_{mtimes n}(mathbb{R}), D>0, and binmathbb{R}^{m}setminus{0}.

Maximal Hörmander Functional Calculus onLpSpaces and UMD Lattices

AbstractLet $A$ be a generator of an analytic semigroup having a Hörmander functional calculus on $X = L^p(\Omega ,Y)$, where $Y$ is a UMD lattice. Using methods from Banach space geometry in connection with functional calculus, we show that for Hörmander spectral multipliers decaying sufficiently fast at $\infty $, there holds a maximal estimate $\| \sup _{t> 0} |m(tA)f|\, \|_{L^p(\Omega ,Y)} \lesssim \|f\|_{L^p(\Omega ,Y)}$. We also show square function estimates $\left \| \left ( \sum _k \sup _{t> 0} |m_k(tA)f_k|^2 \right )^{\frac 12} \right \|_{L^p(\Omega ,Y)} \lesssim \left \| \left ( \sum _k |f_k|^2 \right )^{\frac 12} \right \|_{L^p(\Omega ,Y)}$ for suitable families of spectral multipliers $m_k$, which are even new for the euclidean Laplacian on scalar valued $L^p(\ensuremath {{\mathbb {R}}}^d)$. As corollaries, we obtain maximal estimates for wave propagators and Bochner–Riesz means. Finally, we illustrate the results by giving several examples of operators $A$ that admit a Hörmander functional calculus on some $L^p(\Omega ,Y)$ and discuss examples of lattices $Y$ and non-self-adjoint operators $A$ fitting our context.

A history of solving some famous problems in mathematical analysis

This paper provides a historical view of what functional analysis is, a biography of Per H. Enflo and a short view of four fundamental problems in the geometry of Banach spaces. It gives the story of how Enflo, after several years of work, managed to solve three of these problems. It is a story of trying many ideas, testing them on simpler problems, trying to combine different techniques for handling different aspects of the problems, making many failed attempts and finally succeeding. In 1966 Enflo found a general strategy for solving many problems in infinite dimensions, by combining appropriate finite-dimensional techniques with induction procedures. When applied to the old, fundamental problems in functional analysis, this has led to new concepts, problems, techniques and results in analysis. These developments have been applied also in several other areas of mathematics and computer science.

Existence of Gauss John ellipsoid operator problem

There is a common example in convex geometry and Banach space geometry: the unique ellipsoid with the largest volume associated with each symmetric convex body K is called John ellipsoid. In this paper, we will give some results of John ellipsoid operator problem with Gaussian measure, mainly the continuity of Gaussian measure and the existence of ellipsoid operator.

Адаптивный метод операторной экстраполяции для вариационных неравенств в банаховых пространствах

Многие актуальные задачи исследования операций и математической физики могут быть записаны в форме вариационных неравенств. Разработка и исследование алгоритмов решения вариационных неравенств является направлением прикладного активно развивающегося нелинейного анализа. Отметим, что часто негладкие задачи оптимизации могут эффективно решаться, если переформулировать их в виде седловых задач и применить алгоритмы решения вариационных неравенств. В последнее время наметился прогресс в изучении алгоритмов для задач в банаховых пространствах. Это обусловлено широким привлечением результатов и конструкций геометрии банаховых пространств. В работе предложен и исследован новый алгоритм для разрешения вариационных неравенств в банаховом пространстве. Предлагаемый алгоритм является адаптивным вариантом «forward-reflected-backward algorithm», где используется правило обновления величины шага, не требующее знания лепшицевой константы оператора. Кроме того, вместо метрической проекции на допустимое множество используется обобщенная проекция Альбера. Преимуществом использования алгоритма является лишь одно вычисление на итерационном шаге проекции на допустимое множество. Для вариационных неравенств с монотонными, лепшицевыми операторами, действующими в 2-равномерно выпуклом и равномерно гладком банаховом пространстве, доказана теорема о слабой сходимости метода.

Kadec-Klee property in Musielak-Orlicz function spaces equipped with the Orlicz norm

AbstractIt is well-known that the Kadec-Klee property is an important property in the geometry of Banach spaces. It is closely connected with the approximation compactness and fixed point property of non-expansive mappings. In this paper, a criterion for Musielak-Orlicz function spaces equipped with the Orlicz norm to have the Kadec-Klee property are given. As a corollary, we obtain that a class of non-reflexive Musielak-Orlicz function spaces have the Fixed Point property.

Relative compactness of orbits and geometry of Banach spaces

We investigate for a bounded semigroup of linear operators S on a Banach space E and a vector x∈E, when relative compactness of S(I−T)x for every T∈S implies relative compactness of the orbit Sx. In particular, we derive characterizations of separable Banach spaces not containing c0 and of reflexivity of Banach spaces with a Schauder basis in terms of such compactness results.

On Pointwise ell ^r-Sparse Domination in a Space of Homogeneous Type

We prove a general sparse domination theorem in a space of homogeneous type, in which a vector-valued operator is controlled pointwise by a positive, local expression called a sparse operator. We use the structure of the operator to get sparse domination in which the usual ell ^1-sum in the sparse operator is replaced by an ell ^r-sum. This sparse domination theorem is applicable to various operators from both harmonic analysis and (S)PDE. Using our main theorem, we prove the A_2-theorem for vector-valued Calderón–Zygmund operators in a space of homogeneous type, from which we deduce an anisotropic, mixed-norm Mihlin multiplier theorem. Furthermore, we show quantitative weighted norm inequalities for the Rademacher maximal operator, for which Banach space geometry plays a major role.

In Memoriam: Prof. Roumen P. Maleev, August 17, 1943 - December 16, 2019

Roumen Petrov Maleev was born on August 17, 1943 in the city of Samokov, Bulgaria. He graduated from the Department of Mathematics at Bucurest University in 1967. In 1970 he was appointed as Assistant in the Faculty of Mathematics and Mechanics of Sofia University "St. Kl. Ohridski", where he became Associate Professor in 1983 and Full Professor in 2006.R. Maleev specialized in Moscow State University in the scholarly year 1971/72 and in Warsaw University in 1982 (February--April). He defended his PhD in Sofia University "St. Kl. Ohridski" in 1975 and his DSci dissertation in 1996 (also in Sofia University "St. Kl Ohridski"). During 1989-1995 he served as Deputy Dean of the Faculty of Mathematics and Informatics of at Sofia University "St. Kl. Ohridski''. He has been Head of the Department of Mathematical Analysis of the Faculty (1998--2000) and member of the Specialized Scientific Council on Mathematics and Mechanics (1995--2004).Maleev delivered lecture courses as Visiting Professor in South Florida University in the summer term of 1991 and in the Athens University in May-June 1997. He also presented numerous lectures at various international conferences worldwide.The scientific interests of Prof. Maleev were in the fields of Geometry of Banach spaces, Functional Spaces and Operators, Variational Analysis, Mathematical Analysis, Education in Mathematics and Informatics, Numerical Analysis.

On extreme contractions between real Banach spaces

We completely characterize extreme contractions between two-dimensional strictly convex and smooth real Banach spaces, perhaps for the very first time. In order to obtain the desired characterization, we introduce the notions of (weakly) compatible point pair (CPP) and μ-compatible point pair (μ-CPP) in the geometry of Banach spaces. As a concrete application of our abstract results, we describe all rank one extreme contractions in L(ℓ42,ℓ42) and L(ℓ42,H), where H is any Hilbert space. We also prove that there does not exist any rank one extreme contractions in L(H,ℓp2), whenever p is even and H is any Hilbert space. Finally, we characterize real Hilbert spaces among real Banach spaces in terms of CPP, that substantiates our motivation behind introducing these new geometric notions.

Bishop-Phelps Theorem for Normed Cones

Introduction In the last few years there is a growing interest in the theory of quasi-metric spaces and other related structures such as quasi-normed cones and asymmetric normed linear spaces, because such a theory provides an important tool in the study of several problems in theoretical computer science, approximation theory, applied physics, convex analysis and optimization. Many works on general topology and functional analysis have recently been obtained in order to extend the well-known results of the classical theory of normed linear spaces to the framework of asymmetric normed linear spaces and quasi-normed cones. ‎An abstract cone is analogous to a real vector space‎, ‎except that we take as the set‎ ‎of scalars‎. ‎ In 2004, O‎. ‎Valero introduced the normed cones and proved some closed graph and open mapping results for normed cones. Also Valero defined and studied some properties of quotient normed cones. P. Selinger studied the norm properties of a cone with its order properties and proved Hahn-Banach theorems in these cones under the appropriate conditions. Valero and his colleagues discussed the metrizability of the unit ball of the dual of a normed cone and the isometries of normed cones. Other properties are investigated in a series of papers by Romaguera, Sanchez Perez and Valero. The Bishop-Phelps theorem is a fundamental theorem in functional analysis which has many applications in the geometry of Banach spaces and optimization theory. The classical Bishop-Phelps theorem states that “the set of support functionals for a closed bounded convex subset of a real Banach space X, is norm dense in and the set of support points of is dense in the boundary of ". Indeed, E. Bishop and R. R. Phelps answer a question posed by ‎Victor Klee in 1958. We give an analogue to the normed cones‎, in fact we show that in a continuous normed cone the set of support points of a closed convex set is a dense subset of the boundary under the appropriate hypothesis. Conclusion In this paper the notion of support points of convex sets in normed cones is introduced and it is shown that in a continuous normed cone, under the appropriate conditions, the set of support points of a bounded Scott-closed convex set is nonempty. We also present a Bishop-Phelps type Theorem for normed cones../files/site1/files/52/5.pdf

Type and Cotype Constants and the Linear Stability of Wigner’s Symmetry Theorem

We study the relation between almost-symmetries and the geometry of Banach spaces. We show that any almost-linear extension of a transformation that preserves transition probabilities up to an additive error admits an approximation by a linear map, and the quality of the approximation depends on the type and cotype constants of the involved spaces.

Continuity points and continuous selections of the set-valued metric generalized inverse in Banach spaces

In this paper, continuity points and continuous homogeneous selections of the set-valued metric generalized inverses T∂ in Banach spaces are investigated by means of the methods of geometry of Banach spaces. Moreover, the relation between continuity points and continuous homogeneous selection of the set-valued metric generalized inverse are given in approximatively compact Banach spaces. The results are an answer to the problem posed by Nashed and Votruba.