let $ x, y \in \mathbb{H}. $ For $ \epsilon \in [0, 1), $ we say that $ x $ is $ \epsilon $-orthogonal to $y$, written as $ x \perp^{\epsilon} y, $ if $ |<x, y>| \leq \epsilon \| x \| \| y \|. $ The definition was suitably modified in [@Ca], to obtain an analogous definition of $ \epsilon $-orthogonality in normed linear spaces:

[@Ca] Let $ \mathbb{X} $ be a normed linear space and let $ x, y \in \mathbb{X}. $ For $ \epsilon \in [0, 1)$, we say that $ x $ is $ \epsilon $-orthogonal to $ y $ (in the sense of Birkhoff-James), written as $ x \perp_{D}^{\epsilon} y, $ if $ \| x+\lambda y \|\geq \sqrt[]{1-\epsilon^2}~\|x\|$ for all $\lambda \in \mathbb{R}. $

In [@S] we “decomposed" Birkhoff-James orthogonality (via Definition $ 1.2 $ stated in this paper) in order to obtain a complete characterization of Birkhoff-James orthogonality of bounded linear operators on finite dimensional Banach space. Following similar motivations, in this paper we decompose $ \epsilon $-orthogonality in order to completely characterize Birkhoff-James orthogonality of bounded linear operators, by means of the following definition.

Let $ \mathbb{X} $ be a normed linear space 