The Additive Operator Preserving Birkhoff Orthogonal

Sanxing Guo

doi:10.4236/jamp.2019.73036

Abstract

The Birkhoff orthogonal plays an important role in the geometric study of Banach spaces. It has been con rmed that a Birkhoff orthogonality preserving linear operator between two normed linear spaces must necessarily be a scalar multiple of a linear isometry. In this paper, the author gives a new result that a Birkhoff orthogonality preserving additive operator between two-dimensional normed linear spaces is necessarily a scalar multiple of a linear isometry.

Highlights

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It has been confirmed that a Birkhoff orthogonality preserving linear operator between two normed linear spaces must necessarily be a scalar multiple of a linear isometry
The author gives a new result that a Birkhoff orthogonality preserving additive operator between two-dimensional normed linear spaces is necessarily a scalar multiple of a linear isometry

Summary

Introduction

Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus elit, vestibulum ut, There are several notions of orthogonality in normed linear space, such placerat ac, adipiscing vitae, felis. Duis The Birkhoff orthogonal plays an important role in the geometric nibh mi, congue eu, accumsan eleifend, sagit- study of Banach spaces. Chmielinski found that an orthogonality preserving linear map between two inner product spaces is necessarily a scalar multiple of a linear isometry [8]. This result was extended by [9] to normed linear spaces in the sense of Birkhoff orthogonality with a linear operator. We will show that a Birkhoff orthogonality preserving additive map between two-dimensional normed linear spaces is necessarily a scalar multiple of a linear isometry. The proof is essentially constructive, and some of the techniques may be useful when studying the property of the operator which preserves orthononality

Preliminaries

Main Result