Analytical Study on Approximate ε-Birkhoff-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 of bounded linear operators on a real Banach spaces. Also, using approximate εBirkhoff -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.


Introduction
An impressive growth occurred in the applications of the Birkhoff-James orthogonality that was first introduced in 1935 and used to solve particular problems in the study of geometry of a Banach spaces. In recent times, several authors explored this topic -and obtained many interesting results involving orthogonality of a bounded linear operators. Several recent papers were devoted to the description and classification of the following types of orthogonality in a real normed space .

ISSN: 0067-2904
In this paper, we focus on a specific type of orthogonality which was submitted by Chmielinski . in 2005 and called approximate ---orthogonality. The other types are seen as tools to enable us to resolve some of the outstanding issues in this work. In 2017, Chmielinski et al characterized the approximate ---orthogonality and obtained a sufficient condition for using linear functional on a real normed space . In 2018, Kallol P. et al characterized the approximate --orthogonality of bounded linear operators on a reflexive real Banach space using the norm attainment set. While, a complete characterization of approximate ---orthogonality of bounded linear operators on an infinite dimensional real Banach spaces was obtained . It will also be interesting to conduct an analogous study for an approximate ---orthogonality in other types of spaces such as modular spaces and general fuzzy normed spaces . To proceed in details, we fix some notations and terminologies. Throughout this paper, we will be working with real normed spaces. Let and be the unit ball and unit sphere, respectively, of . Let ( ) denote the set of all bounded (compact) linear operators from to which is the normed space with the supremum norm. In this paper, as an application of the approximate ---orthogonality, we obtain a complete characterization of reflexive Banach spaces in terms of the norm and the minimum norm attainment sets of rank one bounded linear operators on the space.

Preliminaries and set background material
In this section, we recall some concepts and results related to the --orthogonality that will be used in the sequel of approximate ---orthogonality.
The following definition is necessary to obtain the desired characteristics required in this paper. Definition 2.1.
: Let , be two real Banach spaces and . Then: i. is said to be attains norm at a vector in , if . ii. Let denotes the set of all vectors in at which attains norm, i.e., . Apart from those previously reported , some results on the characterization of were also obtained in another work . The set plays an important role in characterizing --orthogonal of bounded linear operators and was obtained in an earlier work . iii. Following similar procedure, the notation of the minimum norm attainment set for is defined in the following way: , where . Also, the set plays a very crucial role in determining the geometry of . In this paper, we obtain a complete characterization of the by applying the concept of approximate ---orthogonality. We further study the relative position of and within this concept. The existence of at least one such vector is guarantee by Hahn Banach Theorem.

Definition 2.2
: For any two vectors and in a normed linear space : i.
is said to be orthogonal to in the sense of Birkhoff-James (for brief, --orthogonality) and written as , if the following is true: for all . ii.
is said to be orthogonal to in the sense of Robert (for brief, -orthogonality) and written as , if the following is true: for all . iii. is said to be approximate -Birkhoff-James orthogonal to in the sense of Chmielinski, (for brief, approximate ---orthogonality) and written as , if the following is true: for all . Otherwise, is not approximate ---orthogonality to and has the symbol . The relation between these concepts and several of their properties can be found in the literature .
Obviously, the relationship between notations is given as follows . Later on, Chmielinski J. , introduced another notion :

Definition 2.3
: For any vector in a normed linear space , a set is said to be an approximate ---orthogonal complement of .
Before going ahead with , we are getting some results which appear to show some sufficient and necessary conditions about reaching the desired result, which states that for every . Let us also note that the following concepts are important in this paper: Definition 2. 4 : A subspace of linear space is said to be a hyperspace, if is a maximal subspace with co-dimension 1. : Let , be two Banach spaces and . Then: i.
is said to be a finite rank operator, if it is a linear operator whose range is finite dimensional (i.e. has a finite dimension). ii. is said to be a compact linear operator, if the image under for any bounded subset of is relatively compact (has compact closure) of . Remark 2.7.
: Any bounded linear operator of a finite-rank is compact.
Among others, there are constructions of a rank one linear operator ( that have one rank, if dimension of ). We connect this concept with several notions to study some properties of a reflexive Banach space . We discuss many of these results and give proofs. Let us finish the introduction with the needed definition. Definition 2. 8 : A normed linear space is a strictly convex, if for any two vectors and in with , then . Theorem 2. 9 : A normed linear space is a strictly convex if and only if for any vector in there exists which attains norm only at vectors of the form with | | . As an application of this definition, we obtain a complete characterization of reflexive Banach spaces in terms of the sets and of the rank one linear operators and the concept of approximate ---orthogonality.
In order to prove the desired results, we make use of the following easy proposition, stated previously . Proposition 2. 10 : For any two vectors and in a normed linear space some properties of approximate ---orthogonality are: i. and for all in and , if and only if .
ii. Note that the relation is homogenous, but neither symmetric nor additive.
iii. For any non-zero vectors and in , if , then and are linearly independent.

Main Results
In this section, we first obtain a complete characterization of and within approximate ---orthogonality. We need the terminology in the following remark to be relevant in this paper.

Remark 3.1: For any two vectors and in
with , and , two subsets of will be defined, that is: , if for all . and , if for all . We will state some obvious but useful properties of this notion which would be used later on in this work, without giving an explicit proof.

Remark 3.4:
The notation signals to the following cases and for any two vectors and in , which we will use in the following theorem that plays an important role in this work.
We strive to obtain a necessary condition for to attain norm at in as the first case.

Reflexivity and rank one bounded linear operator
In this section, we study the sets and of a rank one bounded linear operator on a reflexive Banach space (strictly convex ). As we will observe, this will lead us to an interesting characterization of reflexivity, in terms of these two sets.