On T-orthogonality in Banach spaces

TL;DR

This paper explores the concept of T-orthogonality in Banach spaces, examining its geometric implications, symmetry properties, operator preservation, and its role in characterizing Hilbert spaces.

Contribution

It introduces and analyzes T-orthogonality in Banach spaces, linking it to geometric properties and characterizing Hilbert spaces through this notion.

Findings

T-orthogonality relates to strict convexity, smoothness, and reflexivity.

Characterization of operators preserving T-orthogonality.

Hilbert spaces characterized via T-orthogonality.

Abstract

Let $\mathbb{X}$ be a Banach space and let $\mathbb{X}^*$ be the dual space of $\mathbb{X}.$ For $x,y \in \mathbb{X},$ $ x$ is said to be $T$-orthogonal to $y$ if $Tx(y) =0,$ where $T$ is a bounded linear operator from $\mathbb{X}$ to $\mathbb{X}^*.$ We study the notion of $T$-orthogonality in a Banach space and investigate its relation with the various geometric properties, like strict convexity, smoothness, reflexivity of the space. We explore the notions of left and right symmetric elements w.r.t. the notion of $T$-orthogonality. We characterize bounded linear operators on $\mathbb{X}$ preserving $T$-orthogonality. Finally we characterize Hilbert spaces among all Banach spaces using $T$-orthogonality. \end{abstract}