Dimensional Real Banach Space Research Articles

Multiplicative maps of matrices over rings

Let R be an arbitrary associative ring (not necessarily with an identity). A sufficient condition is given to determine when a multiplicative isomorphism on the matrix ring is additive without the assumption of idempotents. We formulate a version for matrix rings over rings with identities, and then present two applications. One is a new characterization of the automorphisms of the matrix algebra over an arbitrary field by means of multiplicative preservers, and the other one is a new characterization of topologically isomorphisms of finite dimensional real Banach spaces with dimensions no less than 2 by the existence of multiplicative isomorphisms between their rings of all bounded linear operators.

The exponential matrix: an explicit formula by an elementary method

We show an explicit formula, with a quite easy deduction, for the exponential matrix $e^{tA}$ of a real and finite square matrix $A$ (and for complex ones also). The elementary method developed avoids Jordan canonical form, eigenvectors, resolution of any linear system, matrix inversion, polynomial interpolation, complex integration, functional analysis, and generalized Fibonacci sequences. The basic tools are the Cayley-Hamilton theorem and the method of partial fraction decomposition. Two examples are given. We also show that such method applies to algebraic operators on infinite dimensional real Banach spaces.

Wild dynamics and asymptotically separated sets

Let X X be a separable infinite dimensional (real or complex) Banach space. Hájek and Smith, in 2010, constructed a linear bounded operator T T on X X such that A T ≔ { x ∈ X : ‖ T n x ‖ → ∞ } A_T≔\{x\in X:~ \|T^nx\|\to \infty \} is not dense and has nonempty interior, whenever X X admits a symmetric basis. Augé, in 2012, extended the previous construction to each separable Banach space, introducing the notion of wild operators. This work is divided in two parts. In the first part we define and explore the notion of asymptotically separated sets on Banach spaces, giving several examples whenever the ambient space has either finite or infinite dimension. We show how these sets can be used to construct operators with non-intuitive dynamics. Specifically, operators T : X → X T:X\rightarrow X for which the set A T A_T and the set of recurrent points form a partition of the space. Secondly, we investigate geometric and spectral properties of operators with wild dynamic and we provide a wild operator whose spectrum is equal to the closed unit disk. We end this paper giving some comments about the norm closure of the set of wild operators.

Duality of gauges and symplectic forms in vector spaces

A gauge $$\gamma$$ in a vector space X is a distance function given by the Minkowski functional associated to a convex body K containing the origin in its interior. Thus, the outcoming concept of gauge spaces $$(X, \gamma )$$ extends that of finite dimensional real Banach spaces by simply neglecting the symmetry axiom (a viewpoint that Minkowski already had in mind). If the dimension of X is even, then the fixation of a symplectic form yields an identification between X and its dual space $$X^*$$ . The image of the polar body $$K^{\circ }\subseteq X^*$$ under this identification yields a (skew-)dual gauge on X. In this paper, we study geometric properties of this so-called dual gauge, such as its behavior under isometries and its relation to orthogonality. A version of the Mazur–Ulam theorem for gauges is also proved. As an application of the theory, we show that closed characteristics of the boundary of a (smooth) convex body are optimal cases of a certain isoperimetric inequality.

Differential geometry of spatial curves for gauges

We derive Frenet-type results and invariants of spatial curves immersed in 3-dimensional generalized Minkowski spaces, i.e., in linear spaces which satisfy all axioms of finite dimensional real Banach spaces except for the symmetry axiom. Further on, we characterize cylindrical helices and rectifying curves in such spaces, and the computation of invariants is discussed, too. Finally, we study how translations of unit spheres influence invariants of spatial curves.

Banach spaces where convex combinations of relatively weakly open subsets of the unit ball are relatively weakly open

We introduce and study Banach spaces which have property CWO, i.e., every finite convex combination of relatively weakly open subsets of their unit ball is open in the relative weak topology of the unit ball. Stability results of such spaces are established, and we introduce and discuss a geometric condition---property (co)---on a Banach space. Property (co) essentially says that the operation of taking convex combinations of elements of the unit ball is, in a sense, an open map. We show that if a finite dimensional Banach space $X$ has property (co), then for any scattered locally compact Hausdorff space $K$, the space $C_0(K,X)$ of continuous $X$-valued functions vanishing at infinity has property CWO. Several Banach spaces are proved to possess this geometric property; among others: 2-dimensional real spaces, finite dimensional strictly convex spaces, finite dimensional polyhedral spaces, and the complex space $\ell_1^n$. In contrast to this, we provide an example of a $3$-dimensional real Banach space $X$ for which $C_0(K,X)$ fails to have property CWO. We also show that $c_0$-sums of finite dimensional Banach spaces with property (co) have property CWO. In particular, this provides examples of such spaces outside the class of $C_0(K,X)$-spaces.

Topological degree for quasibounded multivalued (S̃)+-perturbations of maximal monotone operators

ABSTRACT Let X be an infinite dimensional real reflexive Banach space with dual space and open and bounded. Let be a maximal monotone operator with and , and let be densely defined strongly quasibounded and of type . A new topological degree theory is introduced for the sum T+C with a degree mapping defined eventually in terms of the Ma degree for multivalued compact operators. Unlike single-valued operators considered by Kartsatos and Skrypnik, the operator C here is multivalued so that the multivalued generalized pseudomonotone operators considered by Browder and Hess include such C and even T+C. Consequently, the main existence results of Browder and Hess are obtained via the new degree theory and some of their existence results are extended. An application of the theory to elliptic partial differential inclusions in divergence form is included.

Geometry of Simplices in Minkowski Spaces

There are many problems and configurations in Euclidean geometry that were never extended to the framework of (normed or) finite dimensional real Banach spaces, although their original versions are inspiring for this type of generalization, and the analogous definitions for normed spaces represent a promising topic. An example is the geometry of simplices in non-Euclidean normed spaces. We present new generalizations of well known properties of Euclidean simplices. These results refer to analogues of circumcenters, Euler lines, and Feuerbach spheres of simplices in normed spaces. Using duality, we also get natural theorems on angular bisectors as well as in- and exspheres of (dual) simplices.

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 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.

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.

Semi-Inner Products and the Concept of Semi-Polarity

The lack of an inner product structure in Banach spaces yields the motivation to introduce a semi-inner product with a more general axiom system, one missing the requirement for symmetry, unlike the one determing a Hilbert space. We use it on a finite dimensional real Banach space $(\X, \| \cdot\|)$ to define and investigate three concepts. First, we generalize that of \emph{antinorms}, already defined in Minkowski planes, for even dimensional spaces. Second, we introduce \emph{normality maps}, which in turn leads us to the study of \emph{semi-polarity}, a variant of the notion of polarity, which makes use of the underlying semi-inner product.

Cross-sections of solution funnels

Let X be a separable infinite dimensional real Banach space. We denote by F(X) the class of continuous functions f:R×X→X such that the ODE u′=f(t,u),u(t0)=x,t0∈R,x∈X, has a global solution for any initial condition. Our main result states that A⊂X is the cross-section of a solution funnel of the ODE u′=f(t,u),u(0)=0, for some f∈F(X), if and only if A is an analytic set.

The image of a closed convex set under a Fredholm operator

The purpose of this article is two-fold. In the first place, we prove that a set is the image of a non empty closed convex subset of a real Banach space under an onto Fredholm operator of positive index if and only if it can be written as the union of {Dn:n∈N}, a non-decreasing family of non empty, closed, convex and bounded sets such that Dn+Dn+2⊆2Dn+1 for every n∈N.The second part of this article proves that in every infinite dimensional real Banach space there is a convex set which can be expressed as the union of countably many closed sets, but not as the union of countably many closed and convex sets. Accordingly, every infinite dimensional real Banach space contains a convex Fσ set which is not the image of a closed convex set under a Fredholm operator.

Reduced Convex Bodies in Finite Dimensional Normed Spaces: A Survey

For a convex body C in a finite dimensional real Banach space M d denote by $${\triangle(C)}$$ its thickness, i.e., its minimal width with respect to the norm. A convex body $${R \subset M^d}$$ is said to be reduced if $${\triangle(C) < \triangle(R)}$$ for each convex body C properly contained in R. The concept of reduced bodies is particularly useful for solving volume-minimizing problems in the area of convexity, and it is also important as extension of basic notions from convexity and functional analysis. Namely, on the one hand the class of reduced bodies is a “dualization” of the concept of complete sets (and, in Euclidean space, that of constant width). On the other hand, it forms a proper superset of the class of complete sets. We present the recent knowledge on this class of convex bodies in finite dimensional real Banach spaces. First we collect general properties of arbitrary reduced bodies. For example, we present constructions supporting our conjecture that in any normed space of dimension larger than 2 there are reduced bodies of unit thickness and diameter at least λ, for every positive number λ. Then we will lay special emphasize on reduced polytopes, and finally on the geometric description of planar reduced bodies. The survey also presents several research problems.

On additive maps preserving certain semi-Fredholm subsets

Let X and Y be two infinite dimensional real or complex Banach spaces, and let φ: ℒ(X) → ℒ(Y) be an additive surjective mapping that preserves semi-Fredholm operators in both directions. In the complex Hilbert space context, Mbekhta and Šemrl [M. Mbekhta and P. Šemrl, Linear maps preserving semi-Fredholm operators and generalized invertibility, Linear Multilinear Algebra 57 (2009), pp. 55–64] determined the structure of the induced map on the Calkin algebra. In this article, we show the following: given an integer n ≥ 1, if φ preserves in both directions ℳ n (X) (resp., 𝒬 n (X)), the set of semi-Fredholm operators on X of non-positive (resp., non-negative) index, having dimension of the kernel (resp., codimension of the range) less than n, then φ(T) = UTV for all T or φ(T) = UT*V for all T, where U and V are two bijective bounded linear, or conjugate linear, mappings between suitable spaces.

Bounded representation and radial projections of bisectors in normed spaces

It is well known that the description of topological and geometric properties of bisectors in normed spaces is a non-trivial subject. In this paper we introduce the concept of bounded representation of bisectors in finite dimensional real Banach spaces. This useful notion com- bines the concepts of bisector and shadow boundary of the unit ball, both corresponding with the same spatial direction. The bounded representation visualizes the connection be- tween the topology of bisectors and shadow boundaries (Lemma 1) and gives the possibility to simplify and to extend some known results on radial projections of bisectors. Our main result (Theorem 1) says that in the manifold case the topology of the closed bisector and the topology of its bounded representation are the same; they are closed, (n − 1)-dimensional balls embedded in Euclidean n-space in the standard way.

On universal covers in normed planes

Abstract In 1914, Lebesgue posed his famous universal cover problem for the Euclidean plane which is still unsolved. We present the first explicit investigation of this problem for arbitrary normed planes. A convex body K in a finite dimensional real Banach space is called universal cover if any set of diameter 1 can be covered by a congruent copy of K; if ”congruent copy“ is replaced by ”translate”, then K is called strong universal cover. We describe the smallest regular hexagons, the smallest equilateral triangles, and the smallest squares (all these figures defined in the sense of the norm) that are strong universal covers in normed planes. In addition, the paper contains a new characterization of Radon planes. In view of universal covers having ball shape we shortly discuss also Jung constants of real Banach spaces, and a known result on Borsuk’s partition problem is reproved.

Minsum hyperspheres in normed spaces

We study the minsum hypersphere problem in finite dimensional real Banach spaces: given a finite set D of (positively weighted) points in an n-dimensional normed space (n≥2), find a minsum hypersphere, i.e., a homothet of the unit sphere of this space that minimizes the sum of (weighted) distances between the hypersphere and the points of D.We show existence results of the following type: there are situations where minsum hyperspheres do not exist, no point-shaped hypersphere can be optimal, and for any norm there exists a set of points D such that a hyperplane is better than any proper hypersphere. We also prove that the intersection of a minsum hypersphere S and conv(D) is non-empty, that D⊆conv(S) implies |S∩conv(D)|≥2, and that |S∩conv(D)|<∞ implies S∩conv(D)⊆D. A certain halving criterion regarding the sums of weights inside and outside of S is verified, and various further results are obtained for large classes of norms, like strictly convex, smooth, and polyhedral norms.

On real analytic functions of unbounded type

In this paper we prove several results on the existence of analytic functions on an infinite dimensional real Banach space which are bounded on some given collection of open sets and unbounded on others. In addition, we also obtain results on the density of some subsets of the space of all analytic functions for natural locally convex topologies on this space.