Approximate Angle in Hilbert $C^{*}$- Modules

Abstract

In this paper, we investigate a type of perturbation of angle in Hilbert $C^{*}$- modules. When a vector $x$ makes approximate angle $\theta$ with a vector $y$, for some $\theta\in[0,\pi]$, we look for another vector $w$, close enough to the vector $y$, so that $x$ makes the exact angle $\theta$ with $w$. This problem has been recently studied for orthogonality by some authors in normed spaces. We have been able to develop the method for angles in the setting of Hilbert $C^{*}$-module. We first consider the problem in inner product spaces, and prove some results in this regard. Then, we try to extend it to some classes of normed spaces such as Hilbert $C^{*}$- modules and $\mathbf{B}(\mathcal{H})$, the algebra of all bounded linear operators on a Hilbert space $\mathcal{H}$.