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

Ali Khalili; Maryam Amyari; Ali Khalili; Maryam Amyari

doi:10.3934/math.2019.3.527

AIMS Mathematics

2019, Volume 4, Issue 3: 527-533. doi: 10.3934/math.2019.3.527

Previous Article Next Article

Research article

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

Ali Khalili ,
Maryam Amyari ^,

Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad 91735, Iran

Received: 10 February 2019 Accepted: 14 May 2019 Published: 29 May 2019
MSC : 46L08, 46L05

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.
- Hilbert $\mathcal{A}$-modules,
- preserving map,
- parallelism,
- $\mathcal{A}$-valued norm parallelism
Citation: Ali Khalili, Maryam Amyari. $\mathcal{A}$-valued norm parallelism in Hilbert $\mathcal{A}$-modules[J]. AIMS Mathematics, 2019, 4(3): 527-533. doi: 10.3934/math.2019.3.527

Related Papers:

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.

References

[1]	L. Arambašića and R. Rajić, On some norm equalities in pre-Hilbert C^modules*, Linear Algebra Appl., 414 (2006), 19-28. doi: 10.1016/j.laa.2005.09.006
[2]	L. Arambasic and R. Rajic, On the C^valued triangle equality and inequality in Hilbert C^-modules, Acta Math. Hung., 119 (2008), 373-380. doi: 10.1007/s10474-007-7055-9
[3]	D. Bakic and B. Guljas, On a class of module maps of Hilbert C^-modules*, Math. Commun., 7 (2002), 177-192.
[4]	T. Bottazzi, C. Conde, M. S. Moslehian, et al. Orthogonality and parallelism of operators on various Banach spaces, J. Aust. Math. Soc., 106 (2019), 160-183. doi: 10.1017/S1446788718000150
[5]	E. C. Lance, Hilbert C^-modules, A Toolkit for Operator Algebraists, Cambridge Univ. Press, Cambridge, 1995.6. M. Mehrazin, M. Amyari, M. Erfanian Omidvar, A new type of Numerical radius of operators on Hilbert C^-modules, Rendiconti del Circolo Matematico di Palermo Series 2, 2018. Available from: http://doi.org/10.1007/s12215-018-0385-3.
[6]	7. A. Zamani, The operator-valued parallelism, Linear Algebra Appl., 505 (201, 282-295. doi: 10.1016/j.laa.2016.05.004
[7]	8. A. Zaman, M. S. Moslehian, Norm-parallelism in the geometry of Hilbert C^-modules*, Indagat. Math. New. Ser., 2(2016), 266-281. doi: 10.1016/j.indag.2015.10.008
[8]	9. A. Zamani, M. S. Moslehian, M. Frank, Angle preserving mappings, Z. Anal. Anwend., 34 (2015), 4-500. doi: 10.4171/ZAA/1551

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)