AIMS Mathematics, 2019, 4(3): 527-533. doi: 10.3934/math.2019.3.527

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

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.