Finite Dimensional Banach Spaces Research Articles

Selections of the metric projection operator and strict solarity of sets with continuous metric projection

In a broad class of finite-dimensional Banach spaces, we show that a closed set with lower semicontinuous metric projection is a strict sun, admits a continuous selection of the metric projection operator onto it, has contractible intersections with balls, and its (nonempty) intersection with any closed ball is a retract of this ball. For sets with continuous metric projection, a number of new results relating the solarity of such sets to the stability of the operator of best approximation are obtained. Bibliography 25 titles.

Ball convex bodies in Minkowski spaces

The notion of ball convexity, considered in finite dimensional real Banach spaces, is a natural and useful extension of usual convexity; one replaces intersections of half-spaces by suitable intersections of balls. A subset $S$ of a normed space is called ball convex if it coincides with its ball hull, which is obtained as intersection of all balls (of fixed radius) containing $S$. Ball convex sets are closely related to notions like ball polytopes, complete sets, bodies of constant width, and spindle convexity. We will study geometric properties of ball convex bodies in normed spaces, for example deriving separation theorems, characterizations of strictly convex norms, and an application to complete sets. Our main results refer to minimal representations of ball convex bodies in terms of their ball exposed faces, to representations of ball hulls of sets via unions of ball hulls of finite subsets, and to ball convexity of increasing unions of ball convex bodies.

On the linear polarization constants of finite dimensional spaces

AbstractWe study the linear polarization constants of finite dimensional Banach spaces. We obtain the correct asymptotic behaviour of these constants for the spaces : they behave as if and as if . For we get the asymptotic behaviour up to a logarithmic factor.

Some remarks on the structure of Lipschitz-free spaces

We give several structural results concerning the Lipschitz-free spaces $\mathcal F(M)$, where $M$ is a metric space. We show that $\mathcal F(M)$ contains a complemented copy of $\ell_1(\Gamma)$, where $\Gamma=\text{dens}(M)$. If $\mathcal N$ is a net in a finite dimensional Banach space $X$, we show that $\mathcal F(\mathcal N)$ is isomorphic to its square. If $X$ contains a complemented copy of $\ell_p, c_0$ then $\f(\mathcal N)$ is isomorphic to its$\ell_1$-sum. Finally, we prove that for all $X\cong C(K)$ spaces, where $K$ is a metrizable compact, $\f(\mathcal N)$ are mutually isomorphic spaces with a Schauder basis.

Norm estimates for random polynomials on the scale of Orlicz spaces

We prove an upper bound for the supremum norm of homogeneous Bernoulli polynomials on the unit ball of finite-dimensional complex Banach spaces. This result is inspired by the famous Kahane–Salem–Zygmund inequality and its recent extensions; in contrast to the known results, our estimates are on the scale of Orlicz spaces instead of ℓp-spaces. Applications are given to multidimensional Bohr radii for holomorphic functions in several complex variables, and to the study of unconditionality of spaces of homogenous polynomials in Banach spaces.

On non-archimedean Gurariĭ spaces

Let UFNA be the class of all non-archimedean finite-dimensional Banach spaces. A non-archimedean Gurariĭ Banach space G over a non-archimedean valued field K is constructed, i.e. a non-archimedean Banach space G of countable type which is of almost universal disposition for the class UFNA. This means: for every isometry g:X→Y, where X,Y∈UFNA and X is a subspace of G, and every ε∈(0,1) there exists an ε-isometry f:Y→G such that f(g(x))=x for all x∈X. We show that all non-archimedean Banach spaces of countable type and of almost universal disposition for the class UFNA are ε-isometric. Furthermore, all non-archimedean Banach spaces of countable type and of almost universal disposition for the class UFNA are isometrically isomorphic if and only if K is spherically complete and {|λ|:λ∈K\\{0}}=(0,∞).

Non-linear plank problems and polynomial inequalities

We study lower bounds for the norm of the product of polynomials and their applications to the so called plank problem. We are particularly interested in polynomials on finite dimensional Banach spaces, in which case our results improve previous works for large numbers of polynomials.

On symmetry of Birkhoff-James orthogonality of linear operators on finite-dimensional real Banach spaces

We characterize left symmetric linear operators on a finite dimensional strictly convex and smooth real normed linear space $ \mathbb{X},$ which answers a question raised recently by one of the authors in \cite{S} [D. Sain, \textit{Birkhoff-James orthogonality of linear operators on finite dimensional Banach spaces, Journal of Mathematical Analysis and Applications, accepted, $ 2016 $}]. We prove that $ T\in B(\mathbb{X}) $ is left symmetric if and only if $ T $ is the zero operator. If $ \mathbb{X} $ is two-dimensional then the same characterization can be obtained without the smoothness assumption. We also explore the properties of right symmetric linear operators defined on a finite dimensional real Banach space. In particular, we prove that smooth linear operators on a finite-dimensional strictly convex and smooth real Banach space can not be right symmetric.

A monotone path-connected set with outer radially lower continuous metric projection is a strict sun

A monotone path-connected set is known to be a sun in a finite-dimensional Banach space. We show that a B-sun (a set whose intersection with each closed ball is a sun or empty) is a sun. We prove that in this event a B-sun with ORL-continuous (outer radially lower continuous) metric projection is a strict sun. This partially converses one well-known result of Brosowski and Deutsch. We also show that a B-solar LG-set (a global minimizer) is a B-connected strict sun.

Angles in normed spaces

The concepts of angle, angle functions, and the question how to measure angles present old and well-established mathematical topics referring to the Euclidean space, and there exist also various extensions to non-Euclidean spaces of different types. In particular, it is very interesting to investigate or to combine (geometric) properties of possible concepts of angle functions and angle measures in finite-dimensional real Banach spaces (= Minkowski spaces). However, going into this direction one will observe that there is no monograph or survey reflecting the complete picture of the existing literature on such concepts in a satisfying manner. We try to close this gap. In this expository paper (containing also new results, and new proofs of known results) the reader will get a comprehensive overview of this field, including further related aspects, as well. For example, angular bisectors, their applications, and angle types which preserve certain kinds of orthogonality are discussed. The latter aspect yields, of course, an interesting link to the large variety of orthogonality types in such spaces.

Birkhoff–James orthogonality of linear operators on finite dimensional Banach spaces

In this paper we characterize Birkhoff–James orthogonality of linear operators defined on a finite dimensional real Banach space X. We also explore the left symmetry of Birkhoff–James orthogonality of linear operators defined on X. Using some of the related results proved in this paper, we finally prove that T∈L(lp2) (p≥2,p≠∞) is left symmetric with respect to Birkhoff–James orthogonality if and only if T is the zero operator.

A separable Fréchet space of almost universal disposition

The Gurariĭ space is the unique separable Banach space G which is of almost universal disposition for finite-dimensional Banach spaces, which means that for every ε>0, for all finite-dimensional normed spaces E⊆F, for every isometric embedding e:E→G there exists an ε-isometric embedding f:F→G such that f↾E=e. We show that GN with a special sequence of semi-norms is of almost universal disposition for finite-dimensional graded Fréchet spaces. The construction relies heavily on the universal operator on the Gurariĭ space, recently constructed by Garbulińska-Węgrzyn and the third author. In addition, we consider a non-graded sequence of semi-norms on GN with which the space GN is of almost universal disposition for finite-dimensional Fréchet spaces with a fixed sequence of semi-norms. In both cases, this yields in particular that GN is universal in the class of all separable Fréchet spaces.

Linear Preservers of Quadratic Operators

Let \({{\mathcal B}(H)}\) be the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space \({H}\). In this paper, we get a complete classification of surjective linear maps on \({{\mathcal B}(H)}\) that preserve quadratic operators in both directions. An analogue result in the setting of finite-dimensional Banach spaces is given.

Syntomic complexes and p-adic nearby cycles

We compute syntomic cohomology of semistable affinoids in terms of cohomology of $$(\varphi ,\Gamma )$$ -modules which, thanks to work of Fontaine–Herr, Andreatta–Iovita, and Kedlaya–Liu, is known to compute Galois cohomology of these affinoids. For a semistable scheme over a mixed characteristic local ring this implies a comparison isomorphism, up to some universal constants, between truncated sheaves of p-adic nearby cycles and syntomic cohomology sheaves. This generalizes the comparison results of Kato, Kurihara, and Tsuji for small Tate twists (where no constants are necessary) as well as the comparison result of Tsuji that holds over the algebraic closure of the field. As an application, we combine this local comparison isomorphism with the theory of finite dimensional Banach Spaces and finiteness of étale cohomology of rigid analytic spaces proved by Scholze to prove a Semistable conjecture for formal schemes with semistable reduction.

Local stability of a generalized Cauchy equation in Banach spaces

Let $${(G, \cdot)}$$ be a group and E be a Banach space. Assume that $${f \colon G\rightarrow E}$$ is a map such that f(G) is an open set containing 0. If there exists an $${\varepsilon > 0}$$ and a p > 1 so that $$\big| \|f(x) + f(y)\| - \|f(xy)\|\big| \leq \varepsilon \min \big \{\|f(x) + f(y)\|^p, \|f(xy)\|^p\big\}$$ for all $${x, y \in G}$$ , then f is an additive map onto E. If E is a finite-dimensional Banach space, the result holds when f(G) (not necessarily open) contains 0 as an interior point.

On subspace-diskcyclicity

In this paper, we define and study subspace-diskcyclic operators. We show that subspace-diskcyclicity does not imply diskcyclicity. We establish a subspace-diskcyclic criterion and use it to find a subspace-diskcyclic operator that is not subspace-hypercyclic for any subspaces. Also, we show that the inverses of invertible subspace-diskcyclic operators do not need to be subspace-diskcyclic for any subspaces. Finally, we prove that every finite-dimensional Banach space over the complex field supports a subspace-diskcyclic operator.

Matrix Factorizations based on induced norms

We decompose a matrix Y into a sum of bilinear terms in a stepwise manner, by considering Y as a mapping between two finite dimensional Banach spaces. We provide transition formulas, and represent them in a duality diagram, thus generalizing the well known duality diagram in the french school of data analysis. As an application, we introduce a family of Euclidean multidimensional scaling models.

Growth Estimates for the Numerical Range of Holomorphic Mappings and Applications

The numerical range of holomorphic mappings arises in many aspects of non-linear analysis, finite and infinite dimensional holomorphy, and complex dynamical systems. In particular, this notion plays a crucial role in establishing exponential and product formulas for semigroups of holomorphic mappings, the study of flow invariance and range conditions, geometric function theory in finite and infinite dimensional Banach spaces, and in the study of complete and semi-complete vector fields and their applications to starlike and spirallike mappings, and to Bloch (univalence) radii for locally biholomorphic mappings. In the present paper, we establish lower and upper bounds for the numerical range of holomorphic mappings in Banach spaces. In addition, we study and discuss some geometric and quantitative analytic aspects of fixed point theory, non-linear resolvents of holomorphic mappings, Bloch radii, as well as radii of starlikeness and spirallikeness.

Sets of unit vectors with small subset sums

We say that a family ${x_i|i\in[m]}$ of vectors in a Banach space $X$ satisfies the $k$-collapsing condition if $|\sum_{i\in I}x_i|\leq 1$ for all $k$-element subsets $I\subseteq{1,2,...,m}$. Let $C(k,d)$ denote the maximum cardinality of a $k$-collapsing family of unit vectors in a $d$\dimensional Banach space, where the maximum is taken over all spaces of dimension $d$. Similarly, let $CB(k,d)$ denote the maximum cardinality if we require in addition that $\sum_{i=1}^m x_i=o$. The case $k=2$ was considered by F\"uredi, Lagarias and Morgan (1991). These conditions originate in a theorem of Lawlor and Morgan (1994) on geometric shortest networks in smooth finite-dimensional Banach spaces. We show that $CB(k,d)=\max{k+1,2d}$ for all $k,d\geq 2$. The behaviour of $C(k,d)$ is not as simple, and we derive various upper and lower bounds for various ranges of $k$ and $d$. These include the exact values $C(k,d)=\max{k+1,2d}$ in certain cases. We use a variety of tools from graph theory, convexity and linear algebra in the proofs: in particular the Hajnal-Szemer\'edi Theorem, the Brunn-Minkowski inequality, and lower bounds for the rank of a perturbation of the identity matrix.

Volume Ratio, Sparsity, and Minimaxity Under Unitarily Invariant Norms

This paper studies non-asymptotic minimax estimation of high-dimensional matrices and provides tight minimax rates for a large collection of loss functions in a variety of problems via information-theoretic methods. Based on the convex geometry of finite-dimensional Banach spaces, we first develop a volume ratio approach for determining minimax estimation rates of unconstrained mean matrices under all unitarily invariant norm losses, which turn out to only depend on the norm of identity matrix. In addition, we establish the minimax rates for estimating normal mean matrices with submatrix sparsity, where the sparsity constraint introduces an additional term in the rate which, in contrast to the unconstrained case, is determined by the smoothness (Lipschitz constant) of the norm. This method is also applicable to the low-rank matrix completion problem and extends well beyond the additive noise model. In particular, it yields tight rates in covariance matrix estimation and Poisson rate matrix estimation problems for all unitarily invariant norms.