From Mazur-Ulam to Wigner

Linear Algebra and its Applications, 2021

Let X and Y be real normed spaces and f : X → Y a surjective mapping. Then f satisfies { f (x) + f (y) , f (x) − f (y) } = { x + y , x − y }, x, y ∈ X, if and only if f is phase equivalent to a surjective linear isometry, that is, f = σU , where U : X → Y is a surjective linear isometry and σ : X → {−1, 1}. This is a Wigner's type result for real normed spaces.

Linear and Multilinear Algebra, 2018

Let X, Y be compact Hausdorff spaces and E, F be Banach spaces over R or C. In this paper, we investigate the general form of surjective (not necessarily linear) isometries T : A −→ B between subspaces A and B of C(X, E) and C(Y, F), respectively. In the case that F is strictly convex, it is shown that there exist a subset Y 0 of Y , a continuous function Φ : Y 0 −→ X onto the set of strong boundary points of A and a family {V y } y∈Y0 of real-linear operators from E to F with V y = 1 such that T f (y) − T 0(y) = V y (f (Φ(y))) (f ∈ A, y ∈ Y 0). In particular, we get some generalizations of the vector-valued Banach-Stone theorem and a generalization of Cambern's result. We also give a similar result in the case that F is not strictly convex, but its unit sphere contains a maximal convex subset which is singleton.

Banach Journal of Mathematical Analysis, 2008

In this paper, we prove the generalized Hyers-Ulam stability of the isometric additive mappings in generalized quasi-Banach spaces, and prove the generalized Hyers-Ulam stability of the isometric additive mappings in generalized p-Banach spaces.

Journal of Mathematical Analysis and Applications, 2001

This paper contains several generalizations of the Mazur-Ulam isometric theorem in F * -spaces which are not assumed to be locally bounded. Let X and Y be two real F * -spaces, and let X be locally pseudoconvex or δ-midpoint bounded. Assume that a operator T maps X onto Y in a δ-locally 1/2 i -isometric manner for all i ∈ 0 ∪ . Then T is affine. In addition, we give the sufficient conditions of a mapping between two topological vector spaces being affine. 

Annals of Functional Analysis, 2017

We demonstrate that any surjective isometry T : A → B not assumed to be linear between unital, completely regular subspaces of complexvalued, continuous functions on compact Hausdorff spaces is of the form T (f) = T (0) + Re µ • (f • τ) + i Im ν • (f • ρ) , where µ and ν are continuous and unimodular, there exists a clopen set K with ν = µ on K and ν = −µ on K c , and τ and ρ are homeomorphisms. T (f) = µ • (f • τ), (1.1) where |µ(y)| = 1 for all y ∈ Y and where τ : Y → X is a homeomorphism. This classic result has been extended to mappings between subspaces of C(X) and C(Y), and a general survey of such results can be found in [4]. We note one in

Dissertationes Mathematicae, 2002

2019

In his 1950 doctoral thesis, [7] Jerison was the first to consider BanachStone problem for isometries on a continuous vector-valued function space. Given an isometry T from C(S,E) to C(Q,E), where S and Q are compact Hausdorff spaces and E is a Banach space, he wanted to know if S and Q were homeomorphic. Jerison showed that the answer is no in general. The idea of a Banach space Y satisfying the Banach-Stone property was first given by Cambern [1]. A pair (E, Y ) of Banach spaces will be said to satisfy the (strong) Banach-Stone property if for every surjective isometry T from C(S,E) to C(Q,Y ), where S,Q are compact Hausdorff spaces, there is a homeomorphism ψ from Q onto S and a map t 7→ h(t) which is continuous from Q into the space B(E, Y ) of bounded operators from E into Y with the strong operator topology such that

Proyecciones (Antofagasta), 2019

Let X be a Banach space and let B(X) be the Banach algebra of all bounded linear operators on X. We characterise surjective (not necessarily linear or additive) maps φ : B(X) → B(X) such that F (φ(A) ¦ φ(B)) = F (A ¦ B) for all A, B ∈ B(X) where F (A) denotes any of R(A) or N (A), and A¦B denotes any binary operations A−B, AB and ABA for all A, B ∈ B(X).

Aequationes Mathematicae, 2000

A description of all continuous (resp. differentiable) solutions f mapping the real line R into a real normed linear space (X, •) (not necessarily strictly convex) of the functional equation f (x + y) = f (x) + f(y) has been presented by Peter Schöpf in [10]. Looking for more readable representations we have shown that any function f of that kind fulfilling merely very mild regularity assumptions has to be proportional to an odd isometry mapping R into X. To gain a proper proof tool we have also established an improvement of Edgar Berz's [4] result on the form of Lebesgue measurable sublinear functionals on R.

We discuss two types of maps on operator spaces. Firstly, through example we show that there is an isometry on unit sphere of an operator space cannot be extended to be a complete isometry on the whole operator space. Secondly, we give a new characterization for complete isometry by the concept of approximate isometry.