$\mathcal{A}$-valued norm parallelism in Hilbert $\mathcal{A}$-modules

Abstract

We define the concept of $\mathcal{A}$-valued norm parallelism in a Hilbert $\mathcal{A}$-module, and then we investigate some properties of this notion and present some characterizations of $\mathcal{A}$-valued norm parallelism in a Hilbert $\mathcal{A}$-module. We also show that if $X$ and $Y$ are two inner product $\mathcal{A}$-modules and $T:X \to Y $ is a linear map such that $|Tx|=|x|$, then $T$ preserves $\mathcal{A}$-valued norm parallelism in both directions.

Keywords

• Hilbert $\mathcal{A}$-modules
• preserving map
• parallelism
• $\mathcal{A}$-valued norm parallelism