[Paper Review] Characterizations of smooth spaces by $ ho_*$-orthogonality

Mohammad Sal Moslehian, Ali Zamani+1 more|arXiv (Cornell University)|May 19, 2017

Advanced Banach Space Theory6 references4 citations

TL;DR

This paper introduces and analyzes $\rho_*^*$-orthogonality in real normed spaces, proving that a linear operator preserving this orthogonality satisfies the norm bounds $\frac{1}{3}\|T\|\|x\| \leq \|Tx\| \leq 3[T]\|x\|$ for all $x \in X$, where $[T] = \inf_{\|x\|=1} \|Tx\|$. It further establishes that $(X, \perp_{\rho_*})$ forms an orthogonality space in the sense of Rätz and provides characterizations of smooth normed spaces via $\rho_*$-orthogonality.

ABSTRACT

The aim of this paper is to present some results concerning the $\ ho_*$-orthogonality in real normed spaces and its preservation by linear operators. Among other things, we prove that if $T\\,: X \\longrightarrow Y$ is a nonzero linear $(I, \ ho_*)$-orthogonality preserving mapping between real normed spaces, then $$\\frac{1}{3}\\|T\\|\\|x\\|\\leq\\|Tx\\|\\leq 3[T]\\|x\\|, \\qquad (x\\in X)$$ where $[T]:=\\inf\\{\\|Tx\\|: \\,x\\in X, \\|x\\|=1\\}$. We also show that the pair $(X,\\perp_{\ ho_*})$ is an orthogonality space in the sense of R\\"{a}tz. Some characterizations of smooth spaces are given based on the $\ ho_*$-orthogonality.

Motivation & Objective

• To investigate the properties of $\rho_*^*$-orthogonality in real normed spaces.
• To study the behavior of linear operators that preserve $\rho_*^*$-orthogonality.
• To establish conditions under which a normed space is smooth using $\rho_*^*$-orthogonality.
• To show that $(X, \perp_{\rho_*})$ satisfies the axioms of an orthogonality space in the sense of Rätz.
• To derive functional equations and stability results related to $\rho_*^*$-orthogonality.

Proposed method

• Defining $\rho_*^*$-orthogonality via the product of left and right norm derivatives: $x \perp_{\rho_*} y \Leftrightarrow \rho_-(x,y)\rho_+(x,y) = 0$.
• Using the norm derivative mappings $\rho_\pm(x,y) = \|x\| \lim_{t \to 0^\pm} \frac{\|x+ty\| - \|x\|}{t}$ to characterize orthogonality.
• Applying the mean value theorem to construct vectors satisfying $x \perp_{\rho_*} y$ and $x+y \perp_{\rho_*} \lambda x - y$ in any 2D subspace.
• Proving that $\rho_*^*$-orthogonality implies the Rätz orthogonality space axioms, ensuring consistency with abstract orthogonality theory.
• Deriving a functional equation characterization: $f(x+y) = f(x)+f(y)$ for $x \perp_{\rho_*} y$ if and only if $f(x) = A(x) + B(x,x)$ with $B(x,y)=0$ when $x \perp_{\rho_*} y$.
• Establishing a Hyers-Ulam-type stability result: if $\|f(x+y)-f(x)-f(y)\| \leq \varepsilon$ for $x \perp_{\rho_*} y$, then $f$ is uniformly close to an additive-quadratic mapping within $\frac{68}{3}\varepsilon$.

Experimental results

Research questions

• RQ1 How does $\rho_*^*$-orthogonality relate to other orthogonality types like Birkhoff–James, isosceles, and semi-inner product orthogonality in normed spaces?
• RQ2 What norm bounds can be established for linear operators that preserve $\rho_*^*$-orthogonality between real normed spaces?
• RQ3 Under what conditions is a normed space smooth, as characterized by $\rho_*^*$-orthogonality?
• RQ4 Does the pair $(X, \perp_{\rho_*})$ satisfy the axioms of an orthogonality space in the sense of Rätz?
• RQ5 Can functional equations involving $\rho_*^*$-orthogonality be decomposed into additive and quadratic components?

Key findings

• For any nonzero linear $(I, \rho_*)$-orthogonality preserving mapping $T: X \to Y$, the norm satisfies $\frac{1}{3}\|T\|\|x\| \leq \|Tx\| \leq 3[T]\|x\|$ for all $x \in X$, where $[T] = \inf_{\|x\|=1} \|Tx\|$.
• The pair $(X, \perp_{\rho_*})$ is an orthogonality space in the sense of Rätz, satisfying the required axioms for such structures.
• In any two-dimensional subspace $P$ of $X$, for every $x \in P$ and $\lambda \geq 0$, there exists $y \in P$ such that $x \perp_{\rho_*} y$ and $x+y \perp_{\rho_*} \lambda x - y$.
• A mapping $f: X \to G$ (with $G$ an Abelian group) satisfies $f(x+y) = f(x)+f(y)$ for all $x \perp_{\rho_*} y$ if and only if $f(x) = A(x) + B(x,x)$ with $B(x,y)=0$ whenever $x \perp_{\rho_*} y$, where $A$ is additive and $B$ is biadditive and symmetric.
• If $f: X \to Y$ (with $Y$ a real Banach space) satisfies $\|f(x+y)-f(x)-f(y)\| \leq \varepsilon$ for all $x \perp_{\rho_*} y$, then $f$ is uniformly within $\frac{68}{3}\varepsilon$ of an additive-quadratic mapping $A(x) + Q(x)$.