Phase-isometries between normed spaces

arXiv (Cornell University), 2020

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.

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

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.

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

Journal of Mathematical Analysis and Applications, 2012

Let X, Y , Z be compact Hausdorff spaces and let E 1 , E 2 , E 3 be Banach spaces. If T : C(X, E 1)×C(Y, E 2) −→ C(Z, E 3) is a bilinear isometry which is stable on constants and E 3 is strictly convex, then there exists a nonempty subset Z 0 of Z, a surjective continuous mapping h : Z 0 −→ X × Y and a continuous function ω : Z 0 −→ Bil(E 1 × E 2 , E 3) such that T (f, g)(z) = ω(z)(f (π X (h(z)), g(π Y (h(z)) for all z ∈ Z 0 and every pair (f, g) ∈ C(X, E 1) × C(Y, E 2). This result generalizes the main theorems in [2] and [6].

Transactions of the American Mathematical Society, 1988

Surjective isometnes between some classical function spaces are investigated. We give a simple technical scheme which verifies whether any such isometry is given by a homeomorphism between corresponding Hausdorff compact spaces. In particular the answer is positive for the C1(X), AC[0,1], LipQ (X) and lipQ (X) spaces provided with various natural norms. 1. Introduction. Let A and B be Banach spaces. By an isometry from A onto

Rocky Mountain Journal of Mathematics, 2000

Let E be a real Banach space. A norm-one element e in E is said to be an isometric reflection vector if there exist a maximal subspace M of E and a linear isometry F : E → E fixing the elements of M and satisfying F (e) = −e. We prove that each of the conditions (i) and (ii) below implies that E is a Hilbert space. (i) There exists a nonrare subset of the unit sphere of E consisting only of isometric reflection vectors, (ii) There is an isometric reflection vector in E, the norm of E is convex transitive, and the identity component of the group of all surjective linear isometries on E relative to the strong operator topology is not reduced to the identity operator on E.

Characterization of isometries in l^2 space, 2022

In general sense, we can define the isometries as transformations which preserve distance between elements. In this paper, we show a characterization of isometries in l^2(X)(for a nonempty set X) which is a Hilbert space .

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.