On mappings approximately preserving orthogonality

We answer a question posed by Chmieliński, whether a linear map which approximately preserves orthogonality must be close to an orthogonality preserving one. Furthermore, we give a short proof of the stability of the orthogonality equation on finite dimensional Hilbert spaces.     

On mappings preserving pentagons

Summary We introduce a new characterization of linear isometries. More precisely, we prove that if a one-to-one mapping f:<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>ℝⁿ→ℝⁿ(2<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>≦n<<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>∞) maps every regular pentagon of side length a> 0 onto a pentagon with side length b> 0, then there exists a linear isometry I :<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>ℝⁿ→ℝⁿup to translation such that f(x) = (b/a) I(x).     

On mappings preserving equilateral triangles

Let E be a Euclidean space, dim E 2. We say that f : E E preserves equilateral triangles if for all triples of points x, y, z E we have We show that if E is a finite-dimensional Euclidean space, dim E 2, f:E E is measurable and preserves equilateral triangles, then it is a similarity transformation (an isometry multiplied by a positive constant). Moreover, in spaces at least three-dimensional we get a similarity transformation without any regularity assumption. Some generalizations as well as some interesting examples are also presented in the paper.     

On strong commutativity preserving mappings

Let R be a ring, and S a non-empty subset of R. Suppose that R admits mappings F and G such that [F(x), G(y)] = [x,y] for all x, y ∈ S. In the present paper, we investigate commutativity of the ring R, when the mapping G is assumed to be a derivation or an endomorphism of R.     

Pythagorean orthogonality and additive mappings

Summary In this note we consider a real normed vector spaceX equipped with the isosceles orthogonality or the Pythagorean orthogonality, both of them defined by R. C. James. It is known that any odd, isosceles orthogonally additive mapping fromX into an Abelian group is unconditionally additive whenever dimX 3. Also, it is worth mentioning that this result was the first of this sort based on a non-homogeneous relation. In this context, we derive here the same for the other non-homogeneous orthogonality, the Pythagorean one, answering in part a pretty old and famous question. The proof uses the corresponding result for isosceles orthogonality and a detailed analysis of the geometry of normed spaces.Dedicated to Professor Jürg Rätz on the occasion of his 60th birthday     

Pythagorean-hodograph preserving mappings

We study the scaled Pythagorean-hodograph (PH) preserving mappings. These mappings make offset-rational isothermal surfaces and map PH curves to PH curves. We present a method to produce a great number of the scaled PH preserving mappings. For an application of the PH preserving mappings, we solve the Hermite interpolation problem for PH curves in the space.     

Mappings approximately preserving orthogonality in normed spaces

We answer many open questions regarding approximately orthogonality preserving mappings (in Birkhoff-James sense) in normed spaces. In particular, we show that every approximately orthogonality preserving linear mapping (in Chmieliński sense) is necessarily a scalar multiple of an ε-isometry. Thus, whenever ε-isometries are close to isometries we obtain stability. An example is given showing that approximately orthogonality preserving mappings are in general far from scalar multiples of isometries, that is, stability does not hold.     

Additive mappings preserving commutativity

The general form of additive surjective mappings on M_nwhich preserve commutativity in both directions is given.     

Stability of disjointness preserving mappings

Let and be compact Hausdorff spaces and let . A linear mapping is called -disjointness preserving if implies that . If is a continuous or surjective -disjointness preserving linear mapping, we prove that there exists a disjointness preserving linear mapping satisfying . We also prove that every unbounded -disjointness preserving linear functional on is disjointness preserving.

     

Holomorphic mappings preserving Minkowski functionals

We show that the equality _m1(f(x))=_m2(g(x))$m_1\left(f\left(x\right)\right)=m_2\left(g\left(x\right)\right)$ for x$x$ in a neighborhood of a point a$a$ remains valid for all x$x$ provided that f$f$ and g$g$ are open holomorphic maps, f(a)=g(a)=0$f\left(a\right)=g\left(a\right)=0$ and _m1,_m2$m_1,m_2$ are Minkowski functionals of bounded balanced domains. Moreover, a polynomial relation between f$f$ and g$g$ is obtained.     

Approximate Roberts orthogonality sets and {(\delta, \varepsilon)}-(a,b)-isosceles-orthogonality preserving mappings

We introduce the concept of approximate Roberts orthogonality set and investigate the geometric properties of such sets. In addition, we introduce the notion of approximate a-isosceles-orthogonality and consider a class of mappings, which approximately preserve a-isosceles-orthogonality.