On -orthogonality in Banach spaces
Volume 172 / 2023
Colloquium Mathematicum 172 (2023), 231-241
MSC: Primary 46B20; Secondary 52A21.
DOI: 10.4064/cm8962-11-2022
Published online: 17 January 2023
Abstract
Let \mathbb X be a Banach space, and let \mathbb X^* be the dual space of \mathbb X and T a bounded linear operator from \mathbb X to \mathbb X^*. For x,y \in \mathbb X, x is said to be T-orthogonal to y if Tx(y) =0. We study the notion of T-orthogonality in a Banach space and investigate its relation to various geometric properties, like strict convexity, smoothness and reflexivity. We explore the notions of left and right symmetric elements with respect to 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.
