Trapezoid Orthogonality in Complex Normed Linear Spaces

Trapezoid Orthogonality in Complex Normed Linear Spaces

Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mi>p</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi...

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Bibliographic Details
Main Authors:	Zheng Li, Tie Zhang, Changjun Li
Format:	Article
Language:	English
Published:	MDPI AG 2025-04-01
Series:	Mathematics
Subjects:	trapezoid orthogonality (T-orthogonality) complex normed linear space complex inner product space Roberts orthogonality Birkhoff orthogonality
Online Access:	https://www.mdpi.com/2227-7390/13/9/1494
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Summary:	Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mi>p</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mo>∥</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>∥</mo></mrow><mi>p</mi></msup><mo>+</mo><msup><mrow><mo>∥</mo><mi>z</mi><mo>∥</mo></mrow><mi>p</mi></msup><msup><mrow><mo>−</mo><mo>∥</mo><mi>x</mi><mo>+</mo><mi>z</mi><mo>∥</mo></mrow><mi>p</mi></msup><mo>−</mo><msup><mrow><mo>∥</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>∥</mo></mrow><mi>p</mi></msup></mrow></semantics></math></inline-formula> be defined on a normed space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">X</mi></semantics></math></inline-formula>. The special case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo><mo>∀</mo><mi>z</mi><mo>∈</mo><mi mathvariant="script">X</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">X</mi></semantics></math></inline-formula> is a real normed linear space, coincides with the trapezoid orthogonality (T-orthogonality), which was originally proposed by Alsina et al. in 1999. In this paper, for the case where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">X</mi></semantics></math></inline-formula> is a complex inner product space endowed with the inner product <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>⟨</mo><mo>·</mo><mo>,</mo><mo>·</mo><mo>⟩</mo></mrow></semantics></math></inline-formula> and induced norm <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∥</mo><mo>·</mo><mo>∥</mo></mrow></semantics></math></inline-formula>, it is proved that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>g</mi><mi>n</mi><mrow><mo>(</mo><msub><mi>G</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><mi>S</mi><mi>g</mi><mi>n</mi><mrow><mo>(</mo><mi>R</mi><mi>e</mi><mrow><mo>⟨</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>⟩</mo></mrow><mo>)</mo></mrow><mo>,</mo><mo>∀</mo><mi>z</mi><mo>∈</mo><mi mathvariant="script">X</mi></mrow></semantics></math></inline-formula>, and a geometric explanation for condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>e</mi><mo>⟨</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>⟩</mo><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> is provided. Furthermore, a condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo><mo>∀</mo><mi>z</mi><mo>∈</mo><mi mathvariant="script">X</mi></mrow></semantics></math></inline-formula> is added to extend the T-orthogonality to the general complex normed linear spaces. Based on some characterizations, the T-orthogonality is compared with several other well-known types of orthogonality. The fact that T-orthogonality implies Roberts orthogonality is also revealed.
ISSN:	2227-7390