On the Generalizations of the Mazur–Ulam Isometric Theorem (original) (raw)
A note on nonlinear isometries between vector-valued function 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.
Some Mappings on Operator Spaces
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.
A Note on Weak Stability of 𝜺-Isometries on Certain Banach Spaces
Journal of Science and Arts
In this paper, we will discuss the weak stability of ε-isometries on certain Banach spaces. Let f: X → Y be a standard ε-isometry. If Y^* is strictly convex, then for any x^*∈X^*, there is φ∈Y^* that satisfies ‖φ‖ ≡r=‖x^* ‖, such that |〈x^*,x〉-〈φ,f(x)〉|≤2rε,x∈X. Also, we show that if X and Y are both L_P spaces (1
Nonlinear isometries between function 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
Isometries of certain operator spaces
2004
Let X and Y be Banach spaces, and L(X, Y ) be the spaces of bounded linear operators from X into Y. In this paper we give full characterization of isometric onto operators of L(X, Y ), for a certain class of Banach spaces, that includes p , 1 < p < ∞. We also characterize the isometric onto operators of L(c 0 ) and K( 1 ), the compact operators on 1 . Furthermore, the multiplicative isometric onto operators of L( 1 ), when multiplication on L( 1 ) is taken to be the Schur product, are characterized.
A Mazur-Ulam theorem in non-Archimedean normed spaces
Nonlinear Analysis: Theory, Methods & …, 2008
The classical Mazur-Ulam theorem which states that every surjective isometry between real normed spaces is affine is not valid for non-Archimedean normed spaces. In this paper, we establish a Mazur-Ulam theorem in the non-Archimedean strictly convex normed spaces.
On Stability of Isometries in Banach Spaces
Springer Optimization and Its Applications, 2011
We analyze the problem of stability of linear isometries (SLI) of Banach spaces. Stability means the existence of a function σ (ε) such that σ (ε) → 0 as ε → 0 and for any ε-isometry A of the space X (i.e., (1 − ε) x ≤ Ax ≤ (1 + ε) x for all x ∈ X) there is an isometry T such that A − T ≤ σ (ε). It is known that all finite-dimensional spaces, Hilbert space, the spaces C(K) and L p (µ) possess the SLI property. We construct examples of Banach spaces X, which have an infinitely smooth norm and are arbitrarily close to the Hilbert space, but fail to possess SLI, even for surjective operators. We also show that there are spaces that have SLI only for surjective operators. To obtain this result we find the functions σ (ε) for the spaces l 1 and l ∞. Finally, we observe some relations between the conditional number of operators and their approximation by operators of similarity.
arXiv Functional Analysis, 2019
For classes of topological vector spaces, we analyze under which conditions open-mapping, bounded-inverse, and closed-graph properties are equivalent. We show that closure under quotients with closed subspaces and closure under closed graphs are sufficient. We show that the class of barreled Pták spaces is exactly the largest class of locally-convex topological vector spaces, which contains all Banach spaces, is closed under quotients with closed subspaces, is closed under closed graphs, is closed under continuous images, and for which an open-mapping theorem, a bounded-inverse theorem, and a closed-graph theorem holds.
A characterization of isometries on an open convex set
Bulletin of the Brazilian Mathematical Society, New Series, 2006
Let X be a real Hilbert space with dim X ≥ 2 and let Y be a real normed space which is strictly convex. In this paper, we generalize a theorem of Benz by proving that if a mapping f , from an open convex subset of X into Y , has a contractive distance ρ and an extensive one Nρ (where N ≥ 2 is a fixed integer), then f is an isometry.
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
Continuous operators on asymmetric normed spaces
Acta Mathematica Hungarica, 2008
If (X, p) and (Y, q) are two asymmetric normed spaces, the set LC(X, Y) of all continuous linear mappings from (X, p) to (Y, q) is not necessarily a linear space, it is a cone. If X and Y are two Banach lattices and p and q are, respectively, their associated asymmetric norms (p(x) = x + , q(y) = y +), we prove that the positive operators from X to Y are elements of the cone LC(X, Y). We also study the dual space of an asymmetric normed space and nally we give open mapping and closed graph type theorems in the framework of asymmetric normed spaces. The classical results for normed spaces follow as particular cases.
Bilinear isometries on spaces of vector-valued continuous functions
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].
Isometries between function spaces
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
2-LOCAL Isometries on Function Spaces
Recent Trends in Operator Theory and Applications, 2019
We study 2-local reflexivity of the set of all surjective isometries between certain function spaces. We do not assume linearity for isometries. We prove that a 2-local isometry in the group of all surjective isometries on the algebra of all continuously differentiable functions on the closed unit interval with respect to several norms is a surjective isometry. We also prove that a 2-local isometry in the group of all surjective isometries on the Banach algebra of all Lipschitz functions on the closed unit interval with the sum-norm is a surjective isometry.
On a characterization of spaces satisfying open mapping and equivalent theorems
arXiv: Functional Analysis, 2019
For classes of topological vector spaces, we analyze under which conditions open-mapping, continuous-inverse, and closed-graph properties are equivalent. Here, closure under quotients with closed subspaces and closure under closed graphs are sufficient. We show that the class of barreled Ptak spaces is exactly the largest class of locally-convex topological vector spaces, which contains all Banach spaces, is closed under quotients with closed subspaces, is closed under closed graphs, is closed under continuous images, and for which an open-mapping theorem, a continuous-inverse theorem, and a closed-graph theorem holds. An analogous, weaker result also holds for the strictly larger class of barreled infra-Ptak spaces.
Isomorphisms of spaces of continuous affine functions
Pacific Journal of Mathematics, 1992
Let K and S be compact convex sets and let A{K) and A(S) be the corresponding Banach spaces of continuous affine functions. If the Banach-Mazur distance between A(K) and A(S) is less than 2, then under certain geometric conditions, the extreme boundaries of K and S are homeomorphic. This extends a result of Amir and Cambern, and has applications to function algebras.