Local Approximate Symmetry of Birkhoff�James Orthogonality in Normed Linear Spaces

c 2021 The Author(s), under exclusive licence to Springer Nature Switzerland AG

1422-6383/21/030001-26 published onlineJune 17, 2021

https://doi.org/10.1007/s00025-021-01437-y Results in Mathematics

Local Approximate Symmetry of

Birkhoff–James Orthogonality in Normed Linear Spaces

Jacek Chmieli´nski, Divya Khurana, and Debmalya Sain

Abstract. Two different notions of approximate Birkhoff–James orthog- onality in normed linear spaces have been introduced by Dragomir and Chmieli´nski. In the present paper we consider a global and a local ap- proximate symmetry of the Birkhoff–James orthogonality related to each of the two definitions. We prove that the considered orthogonality is ap- proximately symmetric in the sense of Dragomir in all finite-dimensional Banach spaces. For the other case, we prove that for finite-dimensional polyhedral Banach spaces, the approximate symmetry of the orthogo- nality is equivalent to some newly introduced geometric property. Our investigations complement and extend the scope of some recent results on a global approximate symmetry of the Birkhoff–James orthogonality.

Mathematics Subject Classification. Primary 46B20; Secondary 51F20, 52B15, 47L05.

Keywords. Birkhoff–James orthogonality, approximate Birkhoff–James orthogonality, C-approximate symmetry, D-approximate symmetry.

1. Introduction

The Birkhoff–James orthogonality is the most natural and well studied notion of orthogonality in normed linear spaces. In general, the Birkhoff–James or- thogonality is not symmetric. Chmieli´nski and W´ojcik [5] introduced a notion of approximate symmetry of the Birkhoff–James orthogonality in normed lin- ear spaces. It should be noted that the authors of [5] considered this notion in the global sense, the meaning of which will be clear once we present the

relevant definition in this section. In this article, our motivation is to consider the corresponding local version of the aforesaid concept. We also study the local version of another standard notion of an approximate Birkhoff–James orthogonality considered in [6]. The advantage of considering the local version is illustrated by obtaining some useful conclusions in the global case, separately for finite-dimensional polyhedral Banach spaces and smooth Banach spaces.

Let us first establish the notations and the terminologies to be used in the present article. Throughout the text, we use the symbolsX, Y to denote real normed linear spaces. Given any two elementsx, y∈X, letxy= conv{x, y}= {(1−t)x+ty:t∈[0,1]} denote the closed line segment joiningxandy. By B_X ={x∈X:x ≤1}andS_X={x∈X :x= 1}we denote the unit ball and the unit sphere ofX, respectively, andB(x, δ) denotes the open unit ball in X centered at xand with the radius δ > 0. The collection of all extreme points ofB_X will be denoted as ExtB_X.

LetX^∗denote the dual space ofX. Given 0=x∈X,f ∈S_X^∗ is said to be asupporting functional atxiff(x) =x. LetJ(x) ={f ∈S_X^∗ :f(x) = x}, 0=x∈X, denote the collection of all supporting functionals atx. Note that for each 0=x∈X, the Hahn-Banach theorem ensures the existence of at least one supporting functional atx.

An elementx∈S_X is said to be a smooth pointifJ(x) ={f} for some f ∈ S_X∗. Let smS_X denote the collection of all smooth points of S_X. In particular if sm S_X =S_X, then X is said to be a smooth space. LetX be a Banach space with a norm . For everyτ >0, the modulus of smoothness is defined by

ρ(τ) = sup

x+τ y+x−τ y −2

2 : x, y∈S_X

. (X, ) is said to be auniformly smooth spaceif lim

τ→0 ρ(τ)

τ = 0.

Let X be a Banach space with a norm . For every ε ∈ (0,2], the modulus of convexity is defined by

δ(ε) = inf

1−x+y

2 : x, y∈B_X, x−y ≥ε

. (X, ) is said to beuniformly convexifδ(ε)>0 for all ε∈(0,2].

It is well known that a Banach space (X, ) is uniformly smooth if and only if its dual space (X^∗, ^∗) is uniformly convex (see [10] for more details).

For x, y ∈ X, we say that x is Birkhoff–James orthogonal to y [2,7], written as x ⊥B y, if x+λy ≥ x for all λ ∈ R. In [7, Theorem 2.1], James proved that if 0=x∈X,y∈X, thenx⊥Byif and only if there exists f ∈J(x) such thatf(y) = 0. We will use the notationsx^⊥={y∈X:x⊥By} and ^⊥x = {y ∈ X : y ⊥B x}. Sain [12] characterized the Birkhoff–James orthogonality of linear operators between finite-dimensional Banach spaces by introducing the notions of the positive part of x, denoted by x⁺, and the negative part of x, denoted by x⁻, for an element x ∈ X. For any element

y∈X, we say that y∈x⁺ (y ∈x⁻) ifx+λy ≥ xfor all λ≥0 (λ≤0).

It is easy to see thatx^⊥=x⁺∩x⁻.

Dragomir [6] defined an approximate Birkhoff–James orthogonality as follows. Let ε ∈ [0,1) and let x, y ∈ X; then x is said to be approximately Birkhoff–James orthogonal toyifx+λy ≥(1−ε)xfor allλ∈R. Later on, Chmieli´nski [3] slightly modified the definition given by Dragomir as follows.

Letε∈[0,1) and letx, y∈X. Thenxis said to be approximately Birkhoff–

James orthogonal toy, written asx⊥^ε_Dy, if and only ifx+λy ≥√

1−ε²x for allλ∈R. Due to this modification, in case of a Hilbert space, the present notion of the approximate orthogonality coincides exactly with the usual no- tion of theε-orthogonality: |x, y| ≤ εxy. In [9, Lemma 3.2], Mal et al.

proved that

x⊥^ε_Dy ⇔ ∃f ∈S_X∗ : |f(x)| ≥

1−ε²x and f(y) = 0. (1.1) Chmieli´nski [3] defined a variation of approximate Birkhoff–James or- thogonality. Given x, y ∈ X and ε ∈ [0,1), x is said to be approximately orthogonal to y, written as x ⊥^ε_B y, if x+λy² ≥ x² −2εxλy for all λ ∈ R. Later, in [4, Theorems 2.2 and 2.3], Chmieli´nski et al. gave two characterizations of this approximate orthogonality:

x⊥^ε_By ⇔ ∃z∈span{x, y}: x⊥B z, and z−y ≤εy; (1.2) x⊥^ε_By ⇔ ∃f ∈J(x) : |f(y)| ≤εy. (1.3) Givenx, y∈X andε∈[0,1), we will write x⊥^ε_D^∗ y (x⊥^ε_B^∗ y) ifx⊥^ε_Dy (x⊥^ε_By) butx⊥^ε_D¹ y (x⊥^ε_B¹y) for any 0≤ε1< ε.

In general, the orthogonality relation between two elements x, y ∈ X need not be symmetric. In other words, for any two elementsx, y∈X,x⊥_B y does not necessarily implyy ⊥_B x. James [8] proved that if dim X ≥3 and the Birkhoff–James orthogonality is symmetric, then the norm is induced by an inner product. For more details on the recent study of these notions of approximate Birkhoff–James orthogonality see [14,15].

In [5], Chmieli´nski and W´ojcik defined the following notion of approxi- mate symmetry of the Birkhoff–James orthogonality in a normed linear space.

Definition 1.1 LetX be a normed linear space. Then the Birkhoff–James or- thogonality is approximately symmetric if there exists ε ∈ [0,1) such that wheneverx, y∈X andx⊥_By, it follows thaty⊥^ε_Bx.

The above definition is global in the sense thatεis independent ofxand y.

In this paper we will work with both of the above mentioned notions of approximate Birkhoff–James orthogonality. To avoid any confusion we will call the above notion of approximate symmetry an approximate symmetry of the Birkhoff–James orthogonality in the sense of Chmieli´nski or shortly:

C-approximate symmetryof the Birkhoff–James orthogonality.

In [5], the authors gave an example of a Banach space where the Birkhoff–

James orthogonality is not C-approximately symmetric. In the present article we will study this example in more detail. The following definition allows us to study local versions of the C-approximate symmetry of the Birkhoff–James orthogonality.

Definition 1.2 LetX be a normed linear space and letx∈X. We say thatx isC-approximately left-symmetric (C-approximately right-symmetric) if there exists ε_x ∈[0,1) such that whenevery ∈X and x⊥B y (y ⊥B x), it follows thaty⊥^ε_B^x x(x⊥^ε_B^x y).

For A ⊆ X we say that the Birkhoff–James orthogonality is C-approximately symmetric on A if there exists ε ∈ [0,1) such that when- everx,y∈ Aandx⊥B y, it follows thaty⊥^ε_Bx.

Let A ⊆ X and let x ∈ S_X. We say that x is C-approximately left- symmetric (C-approximately right-symmetric)onA if there existsε_x∈[0,1) such that whenever y ∈ A and x ⊥B y (y ⊥B x), it follows that y ⊥^ε_B^x x (x⊥^ε_B^xy).

Now, with respect to the Dragomir’s definition, we define the following analogous versions of approximate symmetry considered in Definitions1.1and 1.2.

Definition 1.3 Let X be a normed linear space. We say that the Birkhoff–

James orthogonality is approximately symmetric in the sense of Dragomir, shortly: the Birkhoff–James orthogonality is D-approximately symmetric, if there existsε∈[0,1) such that wheneverx, y∈X andx⊥By, it follows that y ⊥^ε_D x. For x ∈ X, we define xto be D-approximately left-symmetric (D- approximately right-symmetric), if there existsε_x∈[0,1) such that whenever y∈X andx⊥_By (y⊥_Bx), it follows that y⊥^ε_D^x x(x⊥^ε_D^x y).

Observe that we can restrict ourselves to norm-one elements by virtue of the homogeneity of all the notions of orthogonality and approximate orthogo- nality introduced here.

To study the C-approximate left-symmetry and the C-approximate right- symmetry of elements of a normed linear space X, we define the following property. We say that the local property (P) holds forx∈S_X if

x^⊥∩A(x) =∅,

whereA(x) is the collection of all those elementsy∈S_X for which given any f ∈J(y), either f or−f is in J(x).

We say that the property (P) holds for a normed linear spaceX if the local property (P) holds for eachx∈S_X, that is,

for allx∈S_X : the local property (P) holds. (P) IfA ⊆S_X andx∈S_X, then we say that the local property (P) holds for xonAifx^⊥∩A(x)∩ A=∅.

It follows trivially that the local property (P) holds for eachx∈smS_X. We will prove that the local property (P) for anx∈S_X is equivalent to the C-approximate left-symmetry ofxin the local sense, that is, the local property (P) holds forx∈S_X if and only if fory ∈x^⊥∩S_X there exists ε_x,y ∈[0,1) such thaty⊥^ε_B^x,y x.

To study polyhedral Banach spaces, we recall the following definitions from [13] which are relevant to our work:

Definition 1.4 Let X be an n-dimensional Banach space. A polyhedron P is a non-empty compact subset of X which is an intersection of finitely many closed half-spaces of X, that means P = ∩^r_i=1M_i, where M_i are closed half- spaces inX and r∈N. The dimension of the polyhedron P is defined to be the dimension of the subspace generated by the differences x−y of vectors x, y∈P.

Ann-dimensional Banach spaceXis said to be apolyhedral Banach space ifB_X contains only finitely many extreme points, or, equivalently, if S_X is a polyhedron.

Definition 1.5 LetXbe ann-dimensional Banach space. A polyhedronQ⊆X is said to be a face of the polyhedron P ⊆ X if either Q = P or if we can write Q = P ∩δM, where M is a closed half-space in X containing P and δM denotes the boundary ofM. If the dimension ofQ is i, thenQ is called ani-face ofP. (n−1)-faces are calledfacets ofP and 1-faces ofP are called edges ofP.

Definition 1.6 LetX be a finite-dimensional polyhedral Banach space and let Fbe a facet of the unit ballB_X. A functionalf ∈S_X∗ is said to be asupporting functional corresponding to the facetF of the unit ball B_X if the following two conditions are satisfied:

(a) f attains its norm at some pointv ofF. (b) F = (v+ker f)∩S_X.

It is easy to see that there is a unique hyperspaceH such that an affine hyperplane parallel toH contains the facetF of the unit ballB_X. Moreover, there exists a unique norm-one functionalf, such that f attains its norm on F andker f =H. In particular,f is a supporting functional toB_X at every point ofF.

Two elements x, y ∈ ExtB_X of an n-dimensional polyhedral Banach spaceX are said to be adjacentiftx+ (1−t)y= 1 for allt∈[0,1].

Given normed linear spaces X, Y, by B(X, Y) (K(X, Y)) we denote the space of all bounded (compact) linear operators fromXtoY. A bounded linear operatorT ∈ B(X, Y) is said toattain its normatx∈S_X ifT x=T. Let M_T ={x∈S_X :T x=T}be the collection of all norm attaining elements ofT. IfX is a reflexive Banach space andT ∈ K(X, Y), thenM_T =∅(see [1]

for details).

The article is organized as follows. In Sect.2, we study the D-approximate symmetry of the Birkhoff–James orthogonality. For this notion the results we are able to obtain are of the highest level of generality. In particular, we prove that in all finite-dimensional Banach spaces, the Birkhoff–James orthogonality is always D-approximately symmetric.

In Sect.3, we study the C-approximate symmetry of the Birkhoff–James orthogonality. We prove that in finite-dimensional polyhedral Banach spaces, the C-approximate symmetry of the Birkhoff–James orthogonality is equiv- alent to the local property (P) of all elements of ExtB_X. Apparently, the results in this section are less general than in Sect.2. It is caused by the fact that the notion of the C-approximate symmetry essentially differs from the D-approximate one and not all properties remain true. Thus we need to use more subtle methods which usually involve additional assumptions.

In Sect.4, we study the C-approximate symmetry of the Birkhoff–James orthogonality for two-dimensional Banach spaces. Even in this case, establish- ing a satisfactory characterization of the C-approximate symmetry is challeng- ing. To this aim, we introduce a new property, namely property (P1). We show that for any finite-dimensional polyhedral Banach space with property (P1), local property (P) also holds for each element. We also show that the con- verse is true for any two-dimensional polyhedral Banach spaces but in general it need not be true. We show that in a two-dimensional regular polyhedral Banach space with 2nvertices, wheren≥3, the Birkhoff–James orthogonal- ity is C-approximately symmetric. We provide an example to show that the regularity condition in this case cannot be dropped.

2. D-approximate Symmetry of the Birkhoff–James Orthogonality

In [5], Chmieli´nski and W´ojcik proved that in uniformly convex Banach spaces and finite-dimensional smooth Banach spaces, the Birkhoff–James orthogonal- ity is C-approximately symmetric. Our main aim in this section is to prove that for any finite-dimensional Banach space, the Birkhoff–James orthogonality is D-approximately symmetric. To achieve this aim, we first prove the following results.

Theorem 2.1 Let X be a normed linear space and let x, y∈S_X with x⊥^ε_D y for someε∈[0,1) . Then there existε1, ε2>0,ε3∈(0,1) such thatz⊥^ε_D³ w for allz∈B(x, ε1)∩S_X,w∈B(y, ε2)∩S_X.

Proof. Letε1>0 be such that√

1−ε²−ε1>0. Ifz∈B(x, ε1)∩S_X, then z+λy=z−x+x+λy ≥ x+λy − x−z ≥

1−ε²−ε1. Thus for all z ∈ B(x, ε1)∩S_X, we have, z ⊥^δ_D y where δ is such that

√1−ε²−ε1=√ 1−δ².

If|λ| ≥2, then for anyz1, z2∈S_X, we have,z1+λz2 ≥ |λ| −1≥1≥ 1−β for allβ ∈[0,1).

Chooseε2>0 such that√

1−δ²−2ε2>0. Now, ifλ∈Rwith|λ|<2, then for anyz∈B(x, ε1)∩S_X andw∈B(y, ε2)∩S_X, we have,

z+λw=z+λy+λw−λy ≥ z+λy − |λ|y−w>

1−δ²−2ε2. Thus for allz∈B(x, ε1)∩S_X andw∈B(y, ε2)∩S_X, we have,z⊥^ε_D³ wwhere ε3is such that √

1−δ²−2ε2=

1−ε²₃.

Our next result shows that given any two linearly independent elements x, y∈ S_X of a normed linear space X, we can always find an ε∈ [0,1) (de- pending onxandy) such thatx⊥^ε_Dy.

Proposition 2.2 LetXbe a normed linear space and letx, y∈S_Xwithx=±y.

Then there existsε_x,y∈[0,1)such that x⊥^ε_D^x,y y.

Proof. Sincex, y∈S_X and x=±y, it follows thatx,y are linearly indepen- dent. LetX0= span{x, y}and let{x^∗, y^∗} ⊆X₀^∗be such that{x, y;x^∗, y^∗} is a biorthogonal system in X0, where x^∗(x) = y^∗(y) = 1, x^∗(y) =y^∗(x) = 0.

Now, if we takef = _x^x∗_∗, thenf ∈S_X^∗₀,f(x) = _x¹_∗ andf(y) = 0. Let ˆf be a Hahn-Banach extension off toX^∗. Then ˆf ∈S_X^∗, ˆf(x) = _x¹∗ and ˆf(y) = 0.

If _x¹_∗ >1, then for allε∈[0,1), we have, ˆf(x)≥√

1−ε². If _x¹_∗ ≤1, then we can findε ∈[0,1) such that ˆf(x)≥√

1−ε². Thus (1.1) implies that for givenx, y∈S_X, withx=±y, there existsε_x,y ∈[0,1) such that x⊥^ε_D^x,yy.

Theorem 2.3 Let X be a finite-dimensional Banach space. Then the Birkhoff–

James orthogonality is D-approximately symmetric inX.

Proof. Letx∈S_X and lety∈x^⊥∩S_X. Then by Proposition2.2, there exists ε_x,y ∈[0,1) such thaty ⊥^ε_D^x,y x. Letε^∗_x,y be the infimum of all suchε_x,y. We claim that ε := sup

x∈SX

sup

y∈x^⊥∩SX

ε^∗_x,y < 1. If ε = 1, then we can choose {xn}, {y_n} ⊆S_X,ε_n1 such thatx_n⊥By_n andy_n⊥^ε_D^∗n x_n. SinceS_X is compact, there exist convergent sub-sequences of{x_n}, {y_n} which we again denote by {x_n} and {y_n}, respectively. Let x0, y0 ∈ S_X be such that x_n −→ x0 and y_n−→y0. Then by continuity of the norm it follows thaty0∈x^⊥₀ ∩S_X. Now from Proposition 2.2, it follows that y0 ⊥^ε_D⁰ x0 for some ε0 ∈ [0,1). Using Theorem2.1, we can find ε1,ε2>0 andε3∈(0,1) such that w⊥^ε_D³ z for all z ∈ B(x0, ε1)∩S_X and w ∈ B(y0, ε2)∩S_X. Thus we can find m ∈ N such that y_n ⊥^ε_D³ x_n for alln≥m. This leads to a contradiction asy_n ⊥^ε_D^∗n x_n for ε_n1. Thusε <1 and the Birkhoff–James orthogonality is D-approximately

symmetric inX.

Remark 2.4 It follows from the above theorem that each element of a finite- dimensional Banach space is both D-approximately left-symmetric and D- approximately right-symmetric.

We will use the following result from [11] in the proof of the next result.

Theorem 2.5 [11, Theorem 2.1] Let X be a reflexive Banach space and let Y be a normed linear space. Let T, A ∈ K(X, Y) with T = A = 1. Then T ⊥^ε_DA forε∈[0,1) if and only if either(a)or(b) holds.

(a) There exists√ x ∈ M_T such that Ax ∈ (T x)⁺ and for each λ ∈ (−1− 1−ε²,−1 +√

1−ε²), there existsx_λ∈S_X such thatT x_λ+λAx_λ ≥

√1−ε².

(b) There exists√ y ∈ M_T such that Ay ∈ (T y)⁻ and for each λ ∈ (1− 1−ε²,1 +√

1−ε²), there exists y_λ ∈S_X such that T y_λ+λAy_λ ≥

√1−ε².

LetX be a reflexive Banach space and let Y be a normed linear space.

Let T, A ∈ S_K(X,Y) be such that T ⊥_B A. Then by Proposition 2.2, there existsε∈[0,1) such that A⊥^ε_DT. We now estimate the infimum of suchε’s.

Theorem 2.6 Let X be a reflexive Banach space andY a normed linear space.

Suppose that T, A∈ K(X, Y) withT=A= 1and that the set A={x∈ S_X :T x=λAx for allλ∈R} is nonempty. IfA⊥_BT, thenT ⊥^ε_D A, where

√1−ε²= sup_x∈Ainf_λ∈RT x+λAx.

Proof. Letx0 ∈ A. Then T x0 = 0 and by continuity of the functionf(λ) = T x0+λAx0,λ∈Rand the fact thatf(λ)−→ ∞as λ−→ ±∞, it follows that inf_λT x0+λAx0>0. Also, inf_λT x0+λAx0 ≤ T x0 ≤1. Letε_x0 ∈ [0,1) be such that inf_λT x0+λAx0 =

1−ε²_x0. If x∈ X, then it follows from [12, Proposition 2.1] that eitherAx∈(T x)⁺ or Ax∈(T x)⁻. SinceX is a reflexive Banach space and T ∈ K(X, Y), it follows that M_T =∅. Now, by using Theorem2.5, we getT ⊥^ε_D^x0 A. If we fixα= sup_x∈Ainf_λ∈RT x+λAx, then clearlyα∈(0,1]. Letε∈[0,1) be such thatα=√

1−ε². ThenT ⊥^ε_DA

and this completes the proof.

Remark 2.7 The proof of the above theorem suggests that if x0 ∈ A and inf_λT x0+λAx0=

1−ε²_x0, thenT ⊥^ε_D^x0 A. Thus√

1−ε²= sup_x∈Ainf_λ∈R T x+λAxprovides the best possible estimate forε∈[0,1) such thatT ⊥^ε_DA.

As an application of the above theorem, for finite-dimensional spacesX, Y and operatorsT,A∈S_B(X,Y)withA⊥BT, we now obtain an estimate of εsuch thatT ⊥^ε_DA.

Theorem 2.8 LetX, Y be finite-dimensional Banach spaces. LetT, A∈ B(X, Y) with T = A = 1 and let A = {x∈ S_X : T x = λAx for all λ ∈ R}. If A⊥BT, thenT ⊥^ε_DA, where√

1−ε²= sup_x∈Ainf_λ∈RT x+λAx.

Proof. In order to apply Theorem2.6, we need to show that A =∅. Suppose on the contrary thatA=∅. Then for eachx∈S_X, there exists λ_x∈Rsuch that T x =λ_xAx. Clearly, A ⊥B T implies that there does not existλ ∈ R such thatT x=λAx for allx∈X. We now consider the following two cases.

Let rank A ≥2 and let {Ax1, . . . , Ax_k} be a basis for range A, where x1, . . . , x_k∈S_X, 2≤k≤n, where dimX =n. Let{x1, . . . , x_k, x_k+1, . . . , x_n} be a basis forX, where{x_k+1, . . . , x_n} ⊆S_X is a basis for kerA.

In this case there are following two possibilities:

(i) there exist 1≤i, j ≤k such that T x_i =λ_xiAx_i and T x_j =λ_xjAx_j forλ_xi =λ_xj,

(ii) there exists aλ∈Rsuch thatT x_i=λAx_i for each 1≤i≤k.

First consider the case (i). In this case T

x_i+x_j xi+x_j

= λ_xiAx_i+λ_xjAx_j xi+x_j . Using the assumptionA=∅, letλ∈Rbe such that

T

x_i+x_j x_i+x_j

=λA

x_i+x_j x_i+x_j

.

Thus (λ_xi−λ)Ax_i+(λ_xj−λ)Ax_j = 0 and this proves thatλ=λ_xi =λ_xj. This leads to a contradiction asλ_xi=λ_xj.

Now, we will consider the case (ii) as above. In this casex_i0 ∈kerT for at least one i0, k+ 1≤i0 ≤n, otherwise T =λA. Clearly T x_i0 =λAx_i0 for allλ∈R. This contradicts thatA=∅. Thus if rankA≥2, thenA =∅.

Now, consider the case when rank A = 1. Let range A = span {Ax1} wherex1∈S_Xand{x2, . . . , x_n} ⊆S_X be a basis for kerA. By the assumption A= ∅ and thus T x1 =λ_x1Ax1 for some λ_x1 ∈ R. Clearly, A ⊥_B T implies x_i0 ∈ker T for at least one i0, 2≤i0 ≤n. This impliesT x_i0 =λAx_i0 for all λ∈R. This contradicts thatA=∅and thus in this case alsoA =∅. Now, the result follows from Theorem2.6.

The above theorem can be extended to compact operators on a reflexive Banach space, under the additional assumption of injectivity ofAorT. Theorem 2.9 Let X be a reflexive Banach space and Y any normed linear space. Assume that T, A∈ K(X, Y) with T=A = 1 and either A or T is one to one operator. Define A={x∈ S_X : T x =λAx for all λ ∈R}. If A⊥_BT, thenT ⊥^ε_DA, where√

1−ε²= sup_x∈Ainf_λ∈RT x+λAx.

Proof. To prove the result we need to show that A = ∅. Suppose on the contrary that A = ∅. Then for each x∈ S_X, there exists λ_x ∈ R such that T x=λ_xAx. ClearlyA⊥_BT implies that there existx, y∈S_X such that

T x=λ_xAx and T y=λ_yAy (2.1) forλ_x=λ_y. This impliesxandy are linearly independent inX.

Letλ∈Rbe such that T

x+y

x+y

=λA

x+y

x+y

. (2.2)

Also, we have,

T

x+y

x+y

=λ_xAx+λ_yAy

x+y . (2.3)

Let us first assume thatAis one to one operator. Now, using (2.2), (2.3) we haveA((λ_x−λ)x+ (λ_y−λ)y) = 0 and using the assumption thatAis one to one we get (λ_x−λ)x+(λ_y−λ)y= 0. It follows from the linear independence ofx, y that λ_x=λ=λ_y. But this leads to a contradiction as λ_x=λ_y. This implies thatA =∅. Thus in this case the result follows from Theorem2.6.

Now, we assume thatT is one to one operator. It follows from this as- sumption onT that in (2.1)λ_x, λ_y= 0 and also in (2.2), we haveλ= 0. After rewriting (2.2) and using (2.1), we get,

1 λT

x+y

x+y

=A

x+y

x+y

,

1

λxT x+_λ¹

yT y x+y =A

x+y

x+y

. ThusT((_λ¹

x−¹_λ)x+ (_λ¹

y −¹_λ)y) = 0 and using the assumption thatT is one to one we get (_λ¹

x−_λ¹)x+ (_λ¹

y−¹_λ)y= 0. Now, the result follows from the similar arguments as those used in the previous case.

3. C-approximate Symmetry of the Birkhoff–James Orthogonality

It was observed in [5], that in (R², _∞) the Birkhoff–James orthogonality is not C-approximately symmetric. In the following proposition we study the C-approximate left-symmetry and the C-approximate right-symmetry of ele- ments of (Rⁿ, _∞) in detail. In particular, the following result illustrates that in the local sense, the C-approximate left-symmetry is not equivalent to the C-approximate right-symmetry of the Birkhoff–James orthogonality. It is well known that the dual of (Rⁿ, _∞) can be identified with (Rⁿ, 1), where the dual action is given byf(x) =_n

i=1f_ix_ifor all x= (x1, . . . , x_n)∈(Rⁿ, _∞) andf = (f1, . . . , f_n)∈(Rⁿ, 1). Ift∈R, then sgntdenotes the sign function, that is, sgnt= _|t|^t fort= 0 and sgn 0 = 0.

Proposition 3.1 Let X = (Rⁿ, _∞). Then

(i) any smooth point x∈S_X is C-approximately left-symmetric but not C- approximately right-symmetric;

(ii) any extreme point x of S_X is C-approximately right-symmetric but not C-approximately left-symmetric.

Proof. Observe that from the symmetry of S_X, it is sufficient to prove the result for any one of the extreme points and smooth points ofS_X.

(i) Letx = (1, x2, . . . , x_n)∈ S_X be a smooth point. Then |x_i| <1 for all 2 ≤ i ≤ n and J(x) = {f} where f = (1,0, . . . ,0) ∈ S_X∗. Let y = (y1, y2, . . . , y_n) ∈ S_X be such that x ⊥B y. Then by using (1.3), it follows that y1 = 0. As y ∈ S_X, there exists 2 ≤ i0 ≤ n such that |y_i0| = 1. Let g = (0,0, . . . ,0, y_i0,0, . . . ,0) ∈ S_X∗, where y_i0 is the i0-th co-ordinate. Then g ∈ J(y) and |g(x)| = |x_i0| < 1. Thus (1.3) implies that y ⊥^ε_B⁰ x, where ε0=|x_i0|. Now, if we takeε= max

2≤i≤n|x_i|, thenε∈[0,1) andy⊥^ε_B xwhenever x⊥_B y. Hencexis C-approximately left-symmetric.

Now, we show thatxis not C-approximately right-symmetric. If x_i = 0 for all 2≤i ≤n, thenz ⊥_B x, where z = (1,1, . . . ,1). As |f(z)| = 1, there does not exist anyε∈[0,1) such thatx⊥^ε_B z. Without loss of generality we now assume that x2 = 0. Let w = (1,−sgnx2, x3, . . . , x_n) ∈ S_X. Then for any λ ≥ 0 we have w+λx ≥ |1 +λ| ≥ 1. Also, for any λ < 0, we have w+λx ≥ | −sgnx2+λx2|>1.

This shows thatw⊥_B x. As|f(w)|= 1, there does not exist anyε∈[0,1) such that x⊥^ε_B w. Thus, (1.3) implies that x is not C-approximately right- symmetric. Figure1, given below, illustrates this situation forn= 2.

(ii) Considerx= (1,1, . . . ,1) ∈ExtB_X. It follows from the arguments of (i) thatxis not C-approximately left-symmetric.

We now prove thatxis C-approximately right-symmetric. Considery= (y1, y2, . . . , y_n)∈S_X such that y ⊥_B x. Sincey ∈S_X, there exists 1≤i≤n such that|y_i|= 1. Let{i1, i2, . . . , i_k} ⊆ {1,2, . . . , n}be a maximal subset such that|y_ij|= 1 for 1≤j≤k. We now claim thatk >1. Suppose on the contrary thatk= 1. Then y∈smS_X,J(y) ={f}, wheref = (0,0, . . . ,0,1,0, . . . ,0)∈ S_X^∗ and 1 is the i1-th co-ordinate. But f(x) = 0 and this contradicts that y⊥B x.

We now claim that there exist 1≤l =m≤k such thaty_il =−y_im. If y_il =y_im for all 1 ≤l, m≤k, then for sufficiently small absolute value λ, it is easy to see thaty+λx=|yi1+λ|. This clearly contradicts thaty ⊥B x and hence there exist 1≤l =m ≤ksuch that y_il =−yim. Now, if we take g= (0,0, . . . ,0,¹₂,0, . . . ,0,¹₂,0, . . . ,0)∈S_X∗, where ¹₂is ati1-thandi_m-thco- ordinates. Theng∈J(x) andg(y) = 0. This shows thatxis right-symmetric and hence C-approximately right-symmetric. Figure2, given below, illustrates

this situation forn= 2.

Remark 3.2 The proof of the above proposition suggests that inX = (Rⁿ, _∞) the C-approximate right-symmetry (the C-approximate left-symmetry) of any x∈smS_X (x∈ExtB_X) fails because there existsy∈^⊥x∩S_X(y∈x^⊥∩S_X) such that either f ∈ J(y) or −f ∈ J(y) (f ∈ J(x) or −f ∈ J(x)) where J(x) ={f}(f ∈J(y)).

−w z

−z w

g

−g

kerg−x x

y

−y

−f kerf f

Figure 1. Approximate left-symmetry - not approximate right-symmetry.

y x

−x −y g kerg

Figure 2. Approximate right-symmetry - not approximate left-symmetry.

The above remark is the main motivation behind considering the local property (P) forx∈S_X introduced in the first section. Recall that the local property (P) holds forx∈S_X if

x^⊥∩A(x) =∅,

whereA(x) is the collection of all those elementsy∈S_X for which given any f ∈J(y), either f or −f is inJ(x). Also, recall that the property (P) holds for a normed linear spaceX if the local property (P) holds for eachx∈S_X.

We now show that in finite-dimensional Banach spaces, the local property (P) for all elements of ExtB_X implies the property (P) globally for X.

Theorem 3.3 LetX be a finite-dimensional Banach space and suppose that the local property(P)holds for eachx∈ExtB_X. Then the local property(P)holds for eachy∈S_X.

Proof. It follows easily that in any normed linear space, the local property (P) holds for each smooth point. Thus, to prove the result we need to show that the local property (P) holds for any y ∈ S_X\(sm S_X∪ExtB_X). Let y ∈ S_X \(sm S_X ∪ExtB_X). Since S_X is contained in the convex hull of

ExtB_X, letx1, . . . , x_k ∈ExtB_X, k≤ |ExtB_X|, be such thaty =_k

i=1α_ix_i, α_i>0 for all 1≤i≤kand_k

i=1α_i= 1.

Now, we claim that iff ∈J(y), thenf ∈J(x_i) for all 1≤i≤k. Clearly,

|f(x_i)| ≤1 for all 1≤i≤k. Suppose on the contrary thatf(x_j)<1 for some 1≤j≤k. Then

1 =f(y) =

k

i=1

α_if(x_i)<

k

i=1

α_i = 1.

This clearly leads to a contradiction and thus iff ∈J(y), then f ∈J(x_i) for all 1≤i≤k.

Let z ∈ y^⊥∩S_X. Then there exists g ∈ J(y) such that g(z) = 0. But g∈J(x_i) for all 1≤i≤k; this givesx_i⊥_Bz for all 1≤i≤k.

We now claim that there exists some g0 ∈ J(z) such that |g0(y)| < 1.

Suppose on the contrary that for anyg∈J(z) we have|g(y)|= 1, that is, for anyg∈J(z) eithergor−gis inJ(y). Thus for anyg∈J(z) eithergor−g is inJ(x_i) for all 1≤i≤k. This clearly contradicts the local property (P) ofx_i, 1≤i≤k. Thus there exists someg0 ∈J(z) such that|g0(y)|<1 and hence

the local property (P) ofy follows.

The next result shows that the local property (P) ofx∈S_Xis equivalent to the C-approximate left-symmetry ofxin the local sense.

Lemma 3.4 Let X be a normed linear space. Then the local property(P)holds forx∈S_X if and only if fory ∈x^⊥∩S_X, there existsε_x,y ∈[0,1) such that y⊥^ε_B^x,y x.

Proof. We first prove the necessary part of the lemma. Suppose on the contrary that there exists y ∈x^⊥∩S_X such that y ⊥^ε_B xfor any ε ∈ [0,1). Clearly, if f ∈ J(y), then |f(x)| ≤ 1. Thus for all f ∈ J(y) we have |f(x)| = 1 and consequently either f ∈ J(x) or −f ∈ J(x). This contradicts that the local property (P) holds forxand thus the necessary part follows.

We now prove the sufficient part of the lemma. Lety∈x^⊥∩S_X. It follows from the assumption thaty⊥^ε_B^x,yxand, equivalently, there existsf ∈J(y) such that|f(x)| ≤ε_x,y<1, hence±f ∈J(x). Thusy∈A(x) and the local property

(P) ofxfollows.

Observe that in the proof of Theorem 2.1, choosing ε1 = 0 instead of ε1 > 0, we obtain a weaker version of that theorem. The following result is analogous to it.

Lemma 3.5 Let X be a normed linear space and let x, y ∈ S_X with x ⊥^ε_B y for someε∈[0,1). Then there exists δ∈(0,1−ε)such that x⊥^ε_B^+δ z for all z∈B(y, δ)∩S_X.

Proof. Since x ⊥^ε_B y for some ε ∈ [0,1), there exists f ∈ J(x) such that

|f(y)| ≤ε. Now, if we chooseδ∈(0,1−ε), then for allz∈B(y, δ)∩S_X, we have,

|f(z)|=|f(z)−f(y) +f(y)| ≤ |f(y)|+|f(z)−f(y)| ≤ε+δ.

Thusx⊥^ε_B^+δz for allz∈B(y, δ)∩S_X. Lemma3.5says that the C-approximate orthogonality is stable with re- spect to the second vector (small perturbation of it does not cause loss of approximate orthogonality). However, as opposed to D-approximate orthogo- nality (see Theorem 2.1), there is no analogous stability with respect to the first vector. Namely, as it can be observed in the following example, the impli- cation

x⊥^ε_B y ⇒ ∃δ∈(0,1−ε)∀z∈B(x, δ)∩S_X:z⊥^ε_B^+δ y (3.1) need not be true.

Example 3.6 Let X = R² with the maximum norm. Let ε ∈ [0,1) and take x = (1,1), y = (−1,−ε), y0 = (−1,0). Since x⊥By0 and y−y0 = ε, it follows, via (1.2), thatx⊥^ε_By. Assuming that (3.1) is true we take a suitable δ∈(0,1−ε) and setz= (1,1−^δ₂). Thenz∈B(x, δ)∩S_X whencez⊥^ε_By with ε=ε+δ <1. It would mean, again by (1.2), that there existsy ∈S_X such thatz⊥_By andy−y ≤ε<1. However, sincez^⊥∩S_X={(0,1),(0,−1)}, we have y = (0,1) or y = (0,−1) but in both cases y−y ≥ 1 — a contradiction.

We now prove a complete characterization of the C-approximate right- symmetry on any compact subset ofS_X for any normed linear spaceX. Theorem 3.7 Let X be a normed linear space and let A ⊆ S_X be a compact subset. Then anyy∈ A is C-approximately right-symmetric on Aif and only if the local property(P)on Aholds for each x∈ A.

Proof. We first prove the necessary part. Suppose on the contrary thatx∈ A is such thatxfails to have the local property (P) onA. This implies that there existsy ∈x^⊥∩A(x)∩ A. Now, the C-approximate right-symmetry ofy∈ A onAimplies that there existε∈[0,1) andg∈J(y) such that|g(x)| ≤ε. This leads to a contradiction sincey∈A(x) implies|g(x)|= 1.

We now prove the sufficient part. Suppose on the contrary that there existsy∈ Asuch thatyis not C-approximately right-symmetric onA. Observe that if z ∈^⊥y ∩ A, then it follows from similar arguments as those used in Lemma 3.4that there existsε_z,y ∈[0,1) such that y ⊥^ε_B^z,y z. Let ε^∗_z,y be the infimum of all suchε_z,y. By the assumption y is not C-approximately right- symmetric onA, this implies thatε_y = sup_z∈⊥y∩Aε^∗_z,y= 1. Thus we can find {z_n} ⊆^⊥y∩Asuch thaty⊥^ε_B^∗nz_nforε_n 1. Now, from the compactness ofA we can find a convergent subsequence of{z_n}which we again denote by{z_n}. Letz_n−→z0, then by continuity of the norm and compactness ofA, it follows thatz0∈^⊥y∩ A. Again, from similar arguments as those used in Lemma3.4, it follows thaty⊥^ε_B^z0,y z0 for someε_z0,y∈[0,1).