On the generalization of orthogonality and parallelism in $C^{*}-$algebras

Abstract

In this paper, a new norm $\|\cdot\|_\lambda$ is defined on a $C^{*}$-algebra $\mathcal{A}$. Some basic results of $\|\cdot\|_\lambda$ are given. We also give characterizations of parallelism and orthogonality of
operators with respect to $\|\cdot\|_\lambda$. Finally, we obtain some upper bounds of $\|\cdot\|_{\lambda}$.\\