Maliheh Hosseini

Khaje Nasir Toosi University of Technology | KNTU · Faculty of Mathematics

In this paper we deal with the algebraic reflexivity of sets of bounded linear operators on absolutely continuous vector-valued function spaces. As a consequence, it is shown that the set of all surjective linear isometries, the set of all isometric reflections, and the set of all generalized bi-circular projections on AC[0,1] are algebraically ref...

In this paper, we provide a representation of local isometries when defined between certain general subspaces of scalar-valued and vector-valued continuous functions. Based on the description mentioned above, we are able to prove the algebraic reflexivity of the group of isometries of the subspace of absolutely continuous vector-valued functions an...

Let $X$ and $Y$ be compact subsets of $\mathbb{R}$ with at least two points. For $p\geq 1$, let $\AC^p(X)$ be the space of all absolutely continuous complex-valued functions $f$ on $X$ such that $f'\in L^{p}(X)$, with the norm $\left\|f\right\|_{\Sigma}=\left\|f\right\|_\infty+\|f'\|_p$. We describe the topological reflexive closure of the set of l...

Let $X$ and $Y$ be compact subsets of $\mathbb{R}$ with at least two points. For $p\geq 1$, let $\AC^p(X)$ be the space of all absolutely continuous complex-valued functions $f$ on $X$ such that $f'\in L^{p}(X)$, with the norm $\left\|f\right\|_{\Sigma}=\left\|f\right\|_\infty+\|f'\|_p$. We describe the topological reflexive closure of the set of l...

Let K be either the real unit interval [0, 1] or the complex unit circle $${\mathbb {T}}$$ T and let C ( Y ) be the space of all complex-valued continuous functions on a compact Hausdorff space Y . We prove that the isometry group of the algebra $$C^1(K,C(Y))$$ C 1 ( K , C ( Y ) ) of all C ( Y )-valued continuously differentiable maps on K , equipp...

A nonzero projection P on a complex Banach space is called a generalized tricircular projection if there exist distinct modulus one complex numbers λ,μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidema...

Given two subsets X and Y of the real line with at least two points, BV(X) and BV(Y) denote the Banach spaces of all functions of bounded variation on X and Y, respectively. In this paper we study the 2-topological reflexivity of sets of (not necessarily linear) surjective isometries from BV(X) onto BV(Y). In particular, we obtain generalizations o...

In this paper we provide a complete description of projections in the convex hull of two surjective linear isometries (carrying a weighted composition operator form) on absolutely continuous function space AC(X, E), where X is a compact subset of \(\mathbb R\) with at least two points and E is a strictly convex normed space. Among the consequences...

Both classical linear and multilinear isometries defined between subalgebras of bounded continuous functions on (complete) metric spaces are studied. Particularly, we prove that certain such subalgebras, including the subalgebras of uniformly continuous, Lipschitz or locally Lipschitz functions, determine the topology of (complete) metric spaces. A...

Let \(\mathrm {AC}(X)\) be the Banach algebra of all absolutely continuous complex-valued functions f on a compact subset \(X\subset \mathbb {R}\) with at least two points under the norm \(\left\| f\right\| _{\Sigma }=\left\| f\right\| _\infty +\mathrm {V}(f)\), where \(\mathrm {V}(f)\) denotes the total variation of f. We prove that every approxim...

In this paper, we prove that any projection in the convex hull of three surjective linear isometries on AC (X) is a generalized bi-circular projection, where AC (X) denotes the Banach space of all absolutely continuous functions on a compact subset of R with at least two points. We also show that the trivial projections are the only projections on...

For arbitrary subsets X and Y of the real line with at least two points, let B V ( X ) (resp. B V ( Y ) ) be the Banach space of all functions of bounded variation on X (resp. Y) endowed with the natural norm ‖ ⋅ ‖ ∞ + V ( ⋅ ) , where ‖ ⋅ ‖ ∞ and V ( ⋅ ) denote the supremum norm and the total variation of a function, respectively. We show that the...

Let X and Y be compact subsets of R such that X and Y coincide with the closures of their interiors. For any n∈N, let C(n)(X) be the Banach algebra of all n-times continuously differentiable complex-valued functions f on X, with the norm ‖f‖C=maxx∈X⁡(∑k=0n(|f(k)(x)|/k!)). We prove that every approximate local isometry of C(n)(X) to C(n)(Y) is an is...

In this paper we deal with the convergence of sequences of positive linear maps to a (not assumed to be linear) isometry on spaces of continuous functions. We obtain generalizations of known Korovkin-type results and provide several illustrative examples.

Given two subsets X and Y of the real line with at least two points, we apply results on surjective linear isometries between Banach spaces of all functions of bounded variation BV(X) and BV(Y) to show that every 2-local isometry \(T:BV(X)\longrightarrow BV(Y)\) is a constant multiple of an isometric linear algebra isomorphism. Moreover, similar re...

In this paper we deal with surjective linear isometries between spaces of scalar-valued absolutely continuous functions on arbitrary (not necessarily closed or bounded) subsets of the real line (with at least two points). As a corollary, it is shown that when the underlying spaces are connected, each surjective linear isometry of these function spa...

In this paper we study nonlinear diameter preserving mappings defined between function spaces and obtain generalizations of, basically, all known results concerning diameter preservers. In particular, we give a complete description for algebras of continuously differentiable functions, (little) Lipschitz algebras and dense function spaces.

Let G be a locally compact abelian group and B be a commutative Banach algebra. Let (Formula presented.) be the Banach algebra of B-valued Bochner integrable functions on G. In this paper we provide a complete description of continuous disjointness preserving maps on (Formula presented.)-algebras based on a scarcely used tool: the vector-valued Fou...

Let $\alpha_0\in \Bbb C \backslash \{0\}$, $A$ and $B$ be Banach function algebras. Let also $\rho_1:\Omega_1 \longrightarrow A$, $\rho_2:\Omega_2 \longrightarrow A$, $\tau_1:\Omega_1 \longrightarrow B$, and $\tau_2:\Omega_2 \longrightarrow B$ be surjections such that $\|\rho_1(\omega_1)\rho_2(\omega_2)+\alpha_0\|_\infty=\|\tau_1(\omega_1)\tau_2(\o...

This paper investigates the surjective (not necessarily linear) isometries between spaces of absolutely continuous vector-valued functions with respect to the norm ‖⋅‖=max⁡{‖⋅‖∞,V(⋅)}, where ‖⋅‖∞ and V(⋅) denote the supremum norm and the total variation of a function, respectively, and gives an absolutely continuous version of a celebrated theorem...

In this paper we describe, under certain assumptions, surjective diameter preserving mappings when defined between function spaces, not necessarily algebras, thus extending most of the previous results for these operators. We provide an example which shows that our assumptions are not redundant.

Let \(A_1, \ldots , A_k\) be function algebras (or more generally, dense subspaces of uniformly closed function algebras) on locally compact Hausdorff spaces \(X_1, \ldots ,X_k\), respectively, and let Y be a locally compact Hausdorff space. A k-real-linear map \(T:A_1\times \cdots \times A_k\longrightarrow C_0(Y)\) is called a real-multilinear (or...

In this paper we study multilinear isometries defined on certain subspaces of vector-valued continuous functions. We provide conditions under which such maps can be properly represented. Our results contain all known results concerning linear and bilinear isometries defined between spaces of continuous functions. The key result is a vector-valued v...

Let C⁽ⁿ⁾ (I) denote the Banach space of n-times continuously differentiable functions on I = [0, 1], equipped with the norm ||f||n = max { |f(0)|, |fʹ(0)|, … , |f(n–1)(0)|, ||f⁽ⁿ⁾||∞} (f ) ∈ C⁽ⁿ⁾ (I)), where ||·||∞ is the supremum norm. We call a map T : C⁽ⁿ⁾ (I) → C⁽ⁿ⁾(I) a 2-local real-linear isometry if for each pair f, g in C⁽ⁿ⁾(I), there exist...

Let X and Y be subsets of the real line with at least two points. We study the surjective real-linear isometries \({T:BV(X)\longrightarrow BV(Y)}\) between the spaces of functions of bounded variation on X and Y with respect to the supremum norm \({\|\cdot\|_\infty}\) and the complete norm \({\|\cdot\|:=\max(\|\cdot\|_\infty,\mathcal{V}(\cdot))}\),...

Let X, Y be Hausdorff topological spaces, and let E and F be Hausdorff topological vector spaces. For certain subspaces A (X,E) and A(Y, F) of C(X,E) and C(Y, F) respectively (including the spaces of Lipschitz functions), we characterize surjections S, T : A (X;E) → A(Y, F), not assumed to be linear, which jointly preserve common zeros in the sense...

In this paper, we describe into real-linear isometries defined between (not necessarily unital) function algebras and show, based on an example, that this type of isometries behaves differently from surjective real-linear isometries and from classical linear isometries. Next we introduce jointly norm-additive mappings and apply our results on real-...

The main purpose of this paper is to characterize, not necessarily linear, generalized (weakly) peripherally multiplicative maps between Figa-Talamanca Herz algebras. Let G1 and G2 be locally compact Hausdorff groups, Gamma and Omega be arbitrary nonempty sets, and 1 < p < infinity. We characterize surjections S-1 : Gamma -> A(p)(G(1)), S-2 : Omega...

Let be function algebras (or more generally, dense subspaces of uniformly closed function algebras) on locally compact Hausdorff spaces , respectively, and let Z be a locally compact Hausdorff space. A -linear map is called a multilinear (or k-linear) isometry ifBased on a new version of the additive Bishop’s Lemma, we provide a weighted compositio...

The main purpose of this paper is to characterize norm-additive in modulus, not necessarily linear, maps defined between function algebras (not necessarily unital or uniformly closed). In fact, for function algebras A and B on locally compact Hausdorff spaces X and Y, respectively, we study surjections T, S : A -> B satisfying vertical bar vertical...

Let A and B be Banach function algebras on compact Hausdorff spaces X and Y, respectively, and let $\bar A$ and $\bar B$ be their uniform closures. Let I, I′ be arbitrary non-empty sets, α ∈ ℂ\{0}, ρ: I → A, τ: l′ → a and S: I → B T: l′ → B be maps such that ρ(I, τ(I′) and S(I), T(I′) are closed under multiplications and contain exp A and expB,...

Let $A$ and $B$ be subalgebras of $C(X)$ and $C(Y)$, respectively, for some topological spaces $X$ and $Y$. An arbitrary map $T: A\rightarrow B$ is said to be multiplicatively range-preserving if for every $f,g\in A$, $(fg)(X)=(TfTg)(Y)$, and $T$ is said to be separating if $TfTg=0$ whenever $fg=0$. For a given metric space $X$ and $\alpha\in (0,1]...

Let $A$ and $B$ be Banach function algebras on compact Hausdorff spaces $X$ and $Y$, respectively. Given a non-zero scalar $\alpha$and $s,t\in \Bbb N$ we characterize the general form of suitable powers of surjective maps $T, T': A \longrightarrow B$ satisfying $\|(Tf)^s (T'g)^t-\alpha\|_Y=\|f^s g^t-\alpha \|_X$, for all $f,g \in A$, where $\|\cdot...

In this paper we study the behaviour of linear diameter preserving mappings when defined between subalgebras of continuous functions. Namely, we obtain a representation of such mappings as the sum of a weighted compo-sition operator and a linear functional on, at least, the Choquet boundaries of the algebras under consideration. In particular, we g...

Let A and B be Banach function algebras on compact Hausdorff spaces X and Y and let {double pipe}.{double pipe}X and {double pipe}.{double pipe}Y denote the supremum norms on X and Y, respectively. We first establish a result concerning a surjective map T between particular subsets of the uniform closures of A and B, preserving multiplicatively the...

Let A and B be two Banach function algebras on locally compact Hausdorff spaces X and Y, respectively. Let T be a multiplicatively range-preserving map from A onto B in the sense that (TfTg)(Y)=(fg)(X) for all f,g∈A. We define equivalence relations on appropriate subsets X˜ and Y˜ of X and Y, respectively, and show that T induces a homeomorphism be...