Properties of Chmielinski-orthogonality using Kadets-Klee property

Saied Johnny Saied Johnny,Buthainah A A Ahmed Buthainah A A Ahmed

doi:10.52866/ijcsm.2022.01.01.010

Saied Johnny Saied Johnny, Buthainah A A Ahmed Buthainah A A Ahmed

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https://doi.org/10.52866/ijcsm.2022.01.01.010

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Abstract

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.

Highlights

The approximation of BirkhoffJames orthogonality in normed spaces has attracted the attention of mathematicians since its introduction by Chmielinksi, a founder of functional analysis, in 2005 [1]
This paper aims to study the approximate orthogonality (Chmielinski orthogonality) of Birkhoff– James techniques in real Banach space (X, · )—even if the concepts symbolised by ⊥εBJC orthogonality show no ambiguity—and provide some new geometric characterisations that serve as the basis of our main definitions
This paper explores the relation between two types of ⊥εBJC orthogonalities, namely, ⊥εBJC orthogonality in a real Banach space (X, · ) and ⊥εBJC orthogonality in the space of the bounded linear operator B(X, Y)

Summary

Introduction

The approximation of BirkhoffJames orthogonality in normed spaces has attracted the attention of mathematicians since its introduction by Chmielinksi, a founder of functional analysis, in 2005 [1]. We obtain complete characterisations of these two ⊥εBJC orthogonalities in some types of Banach spaces, such as strictly convex, smooth and reflexive spaces. We obtain a smooth compact linear operator with a spectral value that is defined on a reflexive, strictly convex Banach space having a Kadets–Klee property that is either having zero nullity or not ⊥εBJC -right-symmetric.

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