Birkhoff-James Orthogonality and Its Application in the Study of Geometry of Banach Space

We extend to random fields case, the results of Woyczynski, who proved Brunk's type strong law of large numbers (SLLNs) for𝔹-valued random vectors under geometric assumptions. Also, we give probabilistic requirements for above-mentioned SLLN, related to results obtained by Acosta as well as necessary and sufficient probabilistic conditions for the geometry of Banach space associated to the strong and weak law of large numbers for multidimensionally indexed random vectors.

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Finding Minimum Distance on Birkhoff-James Orthogonality in Banach Space

Eksakta Jurnal Ilmu-Ilmu MIPA ◽

10.20885/eksakta.vol1.iss2.art5 ◽

2020 ◽

Vol 20 (2) ◽

pp. 124-128

Author(s):

Susilo Hariyanto ◽

Titi Udjiani ◽

Muhammad Rafid Fadil ◽

Yuri C. Sagala

Keyword(s):

Banach Space ◽

Hilbert Space ◽

Minimum Distance ◽

James Orthogonality

In this paper, we define orthogonality concept on Banach space. This orthogonality is called Birkhoff-James orthogonality. In this paper, we will discuss some new problem about the correlation between orthogonality on Hilbert space and Birkhoff-James orthogonality. Correlation between those two can be investigated by observing the correlation between Hilbert space and Banach space with particular norm. Further, we discuss about the correlation between minimum distance in Banach space with Birkhoff-James orthogonality, by generalized the concept of finding the minimum distance in Hilbert space.

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Analytical Study on Approximate 𝝐-Birkhoff-James Orthogonality

Iraqi Journal of Science ◽

10.24996/ijs.2020.61.7.24 ◽

2020 ◽

pp. 1751-1758

Author(s):

Saied A. Jhonny ◽

Buthainah A. Ahmed

Keyword(s):

Banach Space ◽

Linear Operator ◽

Banach Spaces ◽

Analytical Study ◽

Complete Characterization ◽

Linear Operators ◽

Minimum Norm ◽

Bounded Linear ◽

Rank One ◽

James Orthogonality

In this paper, we obtain a complete characterization for the norm and the minimum norm attainment sets of bounded linear operators on a real Banach spaces at a vector in the unit sphere, using approximate 𝜖-Birkhoff-James orthogonality techniques. As an application of the results, we obtained a useful characterization ofbounded linear operators on a real Banach spaces. Also, using approximate 𝜖-Birkhoff -James orthogonality proved that a Banach space is a reflexive if and only if for any closed hyperspace of , there exists a rank one linear operator such that , for some vectors in and such that 𝜖 .Mathematics subject classification (2010): 46B20, 46B04, 47L05.

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Properties of Chmielinski-orthogonality using Kadets-Klee property

Iraqi Journal for Computer Science and Mathematics ◽

10.52866/ijcsm.2022.01.01.010 ◽

2022 ◽

pp. 94-101

Author(s):

saied Johnny ◽

Buthainah A. A. Ahmed

Keyword(s):

Banach Space ◽

Linear Operator ◽

Real Banach Space ◽

Convex Banach Space ◽

Linear Operators ◽

Bounded Linear ◽

Strictly Convex ◽

Strictly Convex Banach Space ◽

James Orthogonality ◽

New Type

The aim of this paper is to study new results of an approximate orthogonality of Birkhoff-James techniques in real Banach space , namely Chiemelinski orthogonality (even there is no ambiguity between the concepts symbolized by orthogonality) and provide some new geometric characterizations which is considered as the basis of our main definitions. Also, we explore relation between two different types of orthogonalities. First of them orthogonality in a real Banach space and the other orthogonality in the space of bounded linear operator . We obtain a complete characterizations of these two orthogonalities in some types of Banach spaces such as strictly convex space, smooth space and reflexive space. The study is designed to give different results about the concept symmetry of Chmielinski-orthogonality for a compact linear operator on a reflexive, strictly convex Banach space having Kadets-Klee property by exploring a new type of a generalized some results with Birkhoff James orthogonality in the space of bounded linear operators. We also exhibit a smooth compact linear operator with a spectral value that is defined on a reflexive, strictly convex Banach space having Kadets-Klee property either having zero nullity or not -right-symmetric.

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