Approximately bisectrix-orthogonality preserving mappings

Ali Zamani

Commentationes Mathematicae (2014)

  • Volume: 54, Issue: 2
  • ISSN: 2080-1211

Abstract

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Regarding the geometry of a real normed space 𝒳 , we mainly introduce a notion of approximate bisectrix-orthogonality on vectors x , y 𝒳 as follows: x W y if and only if 2 1 - ε 1 + ε x y y x + x y 2 1 + ε 1 - ε x y . We study the class of linear mappings preserving the approximately bisectrix-orthogonality W . In particular, we show that if T : 𝒳 𝒴 is an approximate linear similarity, then x W y T x W T y ( x , y 𝒳 ) for any δ [ 0 , 1 ) and certain θ 0 .

How to cite

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Ali Zamani. "Approximately bisectrix-orthogonality preserving mappings." Commentationes Mathematicae 54.2 (2014): null. <http://eudml.org/doc/292364>.

@article{AliZamani2014,
abstract = {Regarding the geometry of a real normed space $\{\mathcal \{X\}\}$, we mainly introduce a notion of approximate bisectrix-orthogonality on vectors $x, y \in \{\mathcal \{X\}\}$ as follows:\[\{x\mathop \{\mmlmultiscripts\{\perp \mmlprescripts \{\mmlnone \}\{\mmlnone \}\}\}\limits \}\_W y \mbox\{~if and only if~\} \sqrt\{2\}\frac\{1-\varepsilon \}\{1+\varepsilon \}\Vert x\Vert \,\Vert y\Vert \le ~~~ ~~~ \Big \Vert \,\Vert y\Vert x+\Vert x\Vert y\,\Big \Vert \le \sqrt\{2\}\frac\{1+\varepsilon \}\{1-\varepsilon \}\Vert x\Vert \,\Vert y\Vert .\] We study the class of linear mappings preserving the approximately bisectrix-orthogonality $\{\mathop \{\mmlmultiscripts\{\perp \mmlprescripts \{\mmlnone \}\{\mmlnone \}\}\}\limits \}_W$. In particular, we show that if $T: \{\mathcal \{X\}\}\rightarrow \{\mathcal \{Y\}\}$ is an approximate linear similarity, then \[\{x\mathop \{\mmlmultiscripts\{\perp \mmlprescripts \{\mmlnone \}\{\mmlnone \}\}\}\limits \}\_W y\Longrightarrow \{Tx \mathop \{\mmlmultiscripts\{\perp \mmlprescripts \{\mmlnone \}\{\mmlnone \}\}\}\limits \}\_W Ty \qquad (x, y\in \{\mathcal \{X\}\})\] for any $\delta \in [0, 1)$ and certain $\theta \ge 0$.},
author = {Ali Zamani},
journal = {Commentationes Mathematicae},
keywords = {bisectrix-orthogonality; approximate orthogonality; isometry; orthogonality preserving mapping},
language = {eng},
number = {2},
pages = {null},
title = {Approximately bisectrix-orthogonality preserving mappings},
url = {http://eudml.org/doc/292364},
volume = {54},
year = {2014},
}

TY - JOUR
AU - Ali Zamani
TI - Approximately bisectrix-orthogonality preserving mappings
JO - Commentationes Mathematicae
PY - 2014
VL - 54
IS - 2
SP - null
AB - Regarding the geometry of a real normed space ${\mathcal {X}}$, we mainly introduce a notion of approximate bisectrix-orthogonality on vectors $x, y \in {\mathcal {X}}$ as follows:\[{x\mathop {\mmlmultiscripts{\perp \mmlprescripts {\mmlnone }{\mmlnone }}}\limits }_W y \mbox{~if and only if~} \sqrt{2}\frac{1-\varepsilon }{1+\varepsilon }\Vert x\Vert \,\Vert y\Vert \le ~~~ ~~~ \Big \Vert \,\Vert y\Vert x+\Vert x\Vert y\,\Big \Vert \le \sqrt{2}\frac{1+\varepsilon }{1-\varepsilon }\Vert x\Vert \,\Vert y\Vert .\] We study the class of linear mappings preserving the approximately bisectrix-orthogonality ${\mathop {\mmlmultiscripts{\perp \mmlprescripts {\mmlnone }{\mmlnone }}}\limits }_W$. In particular, we show that if $T: {\mathcal {X}}\rightarrow {\mathcal {Y}}$ is an approximate linear similarity, then \[{x\mathop {\mmlmultiscripts{\perp \mmlprescripts {\mmlnone }{\mmlnone }}}\limits }_W y\Longrightarrow {Tx \mathop {\mmlmultiscripts{\perp \mmlprescripts {\mmlnone }{\mmlnone }}}\limits }_W Ty \qquad (x, y\in {\mathcal {X}})\] for any $\delta \in [0, 1)$ and certain $\theta \ge 0$.
LA - eng
KW - bisectrix-orthogonality; approximate orthogonality; isometry; orthogonality preserving mapping
UR - http://eudml.org/doc/292364
ER -

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