On $T$-orthogonality in Banach spaces

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.