- Email: [email protected]
Approximate symmetry of Birkhoff orthogonality
J. Math. Anal. Appl. 461 (2018) 625–640
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Approximate symmetry of Birkhoff orthogonality Jacek Chmieliński ∗ , Paweł Wójcik Department of Mathematics, Pedagogical University of Cracow, Podchor¸ ażych 2, 30-084 Kraków, Poland
a r t i c l e
i n f o
Article history: Received 23 November 2017 Available online 31 January 2018 Submitted by T. Domínguez Benavides
a b s t r a c t In a real normed space X we consider an approximate symmetry of the Birkhoff orthogonality ⊥B and establish its connections with some properties of the space X. Moreover, we introduce and study a new geometric constant for X, connected with the considered property. © 2018 Elsevier Inc. All rights reserved.
Keywords: Birkhoff orthogonality Approximate Birkhoff orthogonality Approximate symmetry of Birkhoff orthogonality
0. Introduction The Birkhoff orthogonality in a normed space is generally not a symmetric relation. However, allowing some inaccuracy one can consider an approximate Birkhoff orthogonality (a new characterization of it has been obtained recently in [8]) and then a related notion of approximate symmetry of this orthogonality can be introduced and investigated. In the present paper we deal with this problem. We consider also some geometrical properties connected with such an approximate symmetry. Throughout the paper we consider a real normed space (X, · ) with dim X ≥ 2. BX and SX stand for the closed unit ball and the unit sphere in X, respectively. For another normed space Y , L(X, Y ) denotes the space of all linear and bounded operators from X into Y . For T ∈ L(X, Y ), by MT we mean the set of unit vectors at which T attains its norm, namely MT := {x ∈ SX : T x = T }. K(X, Y ) is the subspace of L(X, Y ) consisting of all compact operators. X ∗ stands for the dual and X ∗∗ for the bidual space of X. 1. Approximate Birkhoff orthogonality The Birkhoff orthogonality in X is defined by: x⊥B y
⇐⇒
x + λy ≥ x, ∀ λ ∈ R
* Corresponding author. E-mail addresses: [email protected] (J. Chmieliński), [email protected] (P. Wójcik). https://doi.org/10.1016/j.jmaa.2018.01.031 0022-247X/© 2018 Elsevier Inc. All rights reserved.
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
626
(cf. [3,14,15] or a survey [1]). We consider also an approximate Birkhoff orthogonality (ε-Birkhoff orthogonality with ε ∈ [0, 1)): ⇐⇒
x⊥εB y
x + λy2 ≥ x2 − 2εx λy, ∀ λ ∈ R
(1.1)
as introduced in [5]. Obviously, ⊥0B = ⊥B . If X is an inner product space, the approximate orthogonality is naturally defined by: ⇐⇒
x⊥ε y
| x|y | ≤ ε x y,
and in this case ⊥εB coincides with ⊥ε (cf. [5, Proposition 2.1]). In a recent paper [8] the authors have proved the following characterization of the approximate Birkhoff orthogonality. Theorem 1.1 ([8], Theorem 2.3). Let X be a real normed space. For x, y ∈ X and ε ∈ [0, 1): ⇐⇒
x⊥εB y
∃z ∈ Lin{x, y} s.t. x⊥B z, z − y ≤ εy.
(1.2)
Another characterization can be derived from Theorem 1.1 by using the supporting functionals at x ∈ X: J(x) := {ϕ ∈ X ∗ :
ϕ = 1, ϕ(x) = x}.
Theorem 1.2 ([8], Theorem 2.4). Let X be a real normed space. For x, y ∈ X and ε ∈ [0, 1): ⇐⇒
x⊥εB y
∃ ϕ ∈ J(x) s.t. |ϕ(y)| ≤ εy.
(1.3)
Clearly, (1.3) generalizes James’ characterization (cf. [15, Corollary 2.2]): x⊥B y
⇐⇒
∃ϕ ∈ J(x) s.t. ϕ(y) = 0.
(1.4)
A different definition of an approximate Birkhoff orthogonality was given by Dragomir [9]. For a given ε ∈ [0, 1) and x, y ∈ X: x⊥ B y ε
⇐⇒
x + λy ≥ (1 − ε)x, ∀ λ ∈ R.
(1.5)
Some relationships between definitions (1.1) and (1.5) were established in [5] and [19]. In particular, for η inner product spaces ⊥ B is equal to ⊥ with η = 1 − (1 − ε)2 . ε
In an arbitrary normed space Proposition 3.1 in [19] yields x⊥εB y with δ := 1 −
√
⇒
x⊥ B y, δ
1 − 4ε and for ε ≤ 14 . We will improve this result for real spaces.
Theorem 1.3. Let X be a real normed space and let ε ∈ 0, 12 . Then x⊥εB y
⇒
x⊥ B y. 2ε
(1.6)
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
627
Proof. Assume that for x, y ∈ X \ {0} and ε ∈ 0, 12 , we have x⊥εB y. Let z ∈ Lin{x, y} be as in (1.2). The mapping ϕ : R → R, ϕ(t) := x − ty is nonnegative, convex and limt→±∞ ϕ(t) = +∞. Thus it attains its minimum, i.e., there exists β ∈ R such that min{x − αy : α ∈ R} = x − βy. Therefore, for any α ∈ R we get |β| y − x ≤ βy − x ≤ x − αy. Taking α = 0 we get |β| ≤ Now, for an arbitrary α ∈ R we have (1 − 2ε)x = x − 2εx = x −
(1.7) 2x . y
2x εy ≤ x − |β| z − y y
= x − βz − βy ≤ x − βz − βz − βy ≤ (x − βz) + (βz − βy) = x − βy ≤ x − αy, i.e., x⊥ B y. 2ε
2
Yet another connection between the two considered relations was also established in [19]. We will use this result in Section 5. Here δX ∗ denotes the modulus of convexity for the dual space X ∗ . Theorem 1.4 ([19], Proposition 3.8). Let X be a real uniformly smooth normed space and let ε ∈ [0, 2δX ∗ (1)). Then x, y ∈ X, x⊥ B y ε
−1 where η = δX ∗
=⇒
x⊥ηB y,
ε 2 .
2. Approximate symmetry of the Birkhoff orthogonality The Birkhoff orthogonality is, generally, not symmetric (Fig. 1 describes R2 with the maximum norm and vectors x, y such that x⊥B y but y ⊥ B x).
Fig. 1. R2 with the maximum norm; x⊥B y, y ⊥ B x.
If X is a normed space with dim X ≥ 3, then ⊥B is symmetric if and only if X is an inner product space. For 2-dimensional linear spaces it is possible, however, to find a norm (called a Radon norm, cf. [1]) which does not come from an inner product but its Birkhoff orthogonality is symmetric. One can relax a little bit the symmetry requirement and consider an approximate symmetry.
628
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
Definition 2.1. The Birkhoff orthogonality relation in a normed space X is called approximately symmetric, or more precisely: ε-symmetric for some ε ∈ [0, 1), if for any x, y ∈ X: x⊥B y
y⊥εB x.
=⇒
(2.1)
Remark 2.2. If the Birkhoff orthogonality relation is ε-symmetric, then it is also ε -symmetric with an arbitrary ε > ε. On the other hand, if the relation ⊥B is εn -symmetric with εn ∈ [0, 1), n = 1, 2, . . . and εn ε0 , then ⊥B is also ε0 -symmetric. The problem of finding the smallest ε for which the approximate symmetry holds true will be considered in Section 5. Theorem 1.1 yields an immediate characterization of ε-symmetry. Theorem 2.3. The Birkhoff orthogonality in a real normed space X is ε-symmetric for some ε ∈ [0, 1) if and only if for x, y ∈ X: x⊥B y
=⇒
∃ z ∈ Lin{x, y} :
y⊥B z, z − x ≤ εx.
(2.2)
Remark 2.4. Gruber [10] introduced a similar definition: the Birkhoff orthogonality is symmetric up to ε if and only if, for all u, v ∈ SX : u⊥B v
=⇒
∃ w ∈ SX :
v⊥B w, w − u ≤ ε.
(2.3)
It is easy to see that the above definition is equivalent to: x⊥B y
=⇒
∃z ∈ X :
z = x, y⊥B z, z − x ≤ εx
(2.4)
for all x, y ∈ X. Thus the notions of ε-symmetry and of Gruber’s symmetry up to ε practically coincide. Indeed, it is clear that if ⊥B is symmetric up to ε, then it is also ε-symmetric. Conversely, one can show 2ε that if ⊥B is ε-symmetric, then (after replacing z by z := xz z ) it is also symmetric up to ε := 1−ε (hence it is necessary to assume ε < 1/3 in order to keep ε < 1). As we will see, the approximate symmetry of the Birkhoff orthogonality does not imply that the norm comes from an inner product (even if dim X ≥ 3). On the other hand, there are spaces in which the orthogonality relation is not approximately symmetric, i.e., for any ε ∈ [0, 1), ⊥B is not ε-symmetric. Indeed, consider X = R2 with the maximum norm. For x = (1, 1) and y = (0, 1) we have x⊥B y. Suppose that y⊥εB x holds for some ε ∈ [0, 1). It would mean that y⊥B z and z − x ≤ ε for some z. Then z = (α, 0) for some α ∈ R and z − x ≥ 1, a contradiction. See also Corollary 3.6 and Example 3.7. Bearing in mind definition (1.5) and property (1.6), we notice that for ε ∈ 0, 12 condition (2.1) implies x⊥B y
=⇒
y⊥ B x. 2ε
(2.5)
In some cases the above condition may be more convenient for applications and in particular we will use it in Section 5. Symmetry of the approximate orthogonality yields approximate symmetry of the orthogonality. Namely, for an arbitrary normed space X and ε ∈ [0, 1): ⊥εB is symmetric
=⇒
⊥B is ε-symmetric.
(2.6)
Indeed, if x⊥B y, then also x⊥εB y (with an arbitrary ε ∈ [0, 1)) and the assumed symmetry of ⊥εB yields y⊥εB x.
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
629
Let εn ∈ (0, 1), n = 1, 2, . . . and εn → 0 with n → ∞. Then we have: ⊥εBn is symmetric for all n ∈ N
=⇒
⊥B is symmetric.
(2.7)
Indeed, assume x⊥B y; then, for n ∈ N, also x⊥εBn y and it follows from the assumption that y⊥εBn x. Hence there exist zn ∈ Lin{x, y} such that x − zn ≤ εn x and y⊥B zn (n ∈ N). Letting n → ∞, we get zn → x and y⊥B x. On the other hand, symmetry of the Birkhoff orthogonality does not imply symmetry of the approximate orthogonality. Example 2.5. Let X = R2 with the l∞ -l1 -norm: max{|x|, |y|}, (x, y) = |x| + |y|,
if xy ≥ 0; if xy < 0.
The above norm is a Radon norm, i.e., the Birkhoff orthogonality relation is symmetric. (See Fig. 2.)
Fig. 2. R2 with (a Radon) l∞ -l1 -norm; x⊥εB y, y ⊥εB x.
Let x = (1, 0), y = (ε, ε − 1) with some arbitrarily fixed ε ∈ (0, 1). For z = (0, −1) we have x⊥B z and y − z = (ε, ε) = ε = εz, whence x⊥εB y. Suppose that y⊥εB x. Then it would exist w such that y⊥B w and w − x ≤ εx = ε. If y⊥B w, then w must be of the form w = (t, t) with some t ∈ R. Then w − x = (t − 1, t) ≥ 1 > ε, — a contradiction. Hence y⊥εB x. 3. Geometrical properties related to the approximate symmetry We recall the notion of norm derivatives (cf. [2]): x + ty2 − x2 2t t→0± x + ty − x = x · lim , t t→0±
ρ± (x, y) := lim
∀ x, y ∈ X, t ∈ R.
For each x ∈ X, ρ± (x, x) = x2 . Generally, ρ− ≤ ρ+ . If x0 ∈ X \ {0} is a point of smoothness, then the norm is Gateaux differentiable at x0 (cf. [2, Definition 2.1.1]) whereas ρ− (x0 , y) = ρ+ (x0 , y), In particular, if X is smooth, then ρ− and ρ+ coincide.
∀ y ∈ X.
(3.1)
630
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
The mappings ρ+ and ρ− are subadditive and superadditive, respectively, in the second variable (cf. [2, Proposition 2.1.3]). It follows from (3.1) that for x0 ∈ X \ {0} being a point of smoothness we have ρ± (x0 , y + z) = ρ± (x0 , y) + ρ± (x0 , z),
∀ y, z ∈ X.
(3.2)
In particular, for smooth spaces, both mappings ρ± are additive in the second variable. Norm derivatives yield orthogonality relations: x⊥ρ± y
⇐⇒
ρ± (x, y) = 0,
and related to them approximate orthogonalities: x⊥ερ± y
⇐⇒
|ρ± (x, y)| ≤ εx y.
It is known that ⊥ερ± ⊂ ⊥εB (cf. [7, Theorem 3.2]). The reverse inclusion is, generally, not true unless some kind of smoothness is assumed. Proposition 3.1. If the norm is smooth at x0 ∈ X \ {0}, then x0 ⊥εB y =⇒ x0 ⊥ερ± y,
∀ y ∈ X.
(3.3)
In particular, the equality ⊥ερ± = ⊥εB holds true if X is smooth. Proof. Assuming x0 ⊥εB y, we have (cf. [7, Theorem 3.1]) ρ− (x0 , y) − εx0 y ≤ 0 ≤ ρ+ (x0 , y) + εx0 y. Since X is smooth at x0 , we have ρ− (x0 , ·) = ρ+ (x0 , ·) whence ρ± (x0 , y) − εx0 y ≤ 0 ≤ ρ± (x0 , y) + εx0 y, and hence |ρ± (x0 , y)| ≤ εx0 y, i.e., x0 ⊥ερ± y.
2
For a normed space X we define (cf. [22]) a constant R(X) := sup{x − y : conv {x, y} ⊂ SX }. We will also consider the following property of X which is essentially weaker than smoothness or strict convexity of X: x, y ∈ X, x = y, conv {x, y} ⊂ SX =⇒ X is smooth at x − y.
(3.4)
Obviously, smooth spaces as well as strictly convex ones satisfy (3.4). On the other hand, it is easy to see that R2 with the maximum norm, which is neither strictly convex nor smooth, satisfies (3.4). Actually, one can consider X = R2 with any norm for which the unit ball is a symmetric polygon such that its sides are not parallel to diagonals. Then the condition (3.4) is satisfied. Notice that the property (3.4) is hereditary, i.e., if X satisfies this property then so does any subspace of X. Theorem 3.3 below gives a large family of spaces satisfying (3.4). In the proof of this theorem we will apply a characterization of points of smoothness in the space of compact operators K(X, Y ).
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
631
Theorem 3.2 ([13], Corollary 2.2). Let X, Y be Banach spaces and let X be reflexive. Then a compact operator T is smooth in K(X, Y ) if and only if there exists x0 ∈ SX such that MT = {x0 , −x0 } and T x0 is smooth in Y . Theorem 3.3. Let X be a two-dimensional strictly convex normed space and let Y be a strictly convex and smooth normed space. Then L(X, Y ) (and each its subspace) satisfies the condition (3.4). Proof. Obviously, in our case K(X, Y ) = L(X, Y ). Let A, B ∈ L(X, Y ) such that A = B, A = B = 1 and conv{A, B} ⊂ SL(X,Y ) be fixed. Then A +B = A +B. It follows from [23, Theorem 2.1] that there exists a vector x0 ∈ SX such that Ax0 = A, Bx0 = B, Ax0 + Bx0 = Ax0 + Bx0 . Therefore Ax +Bx 0 0 = 1. Strict convexity of Y yields Ax0 = Bx0 . Thus x0 ∈ ker(A − B) and since dim X = 2, it 2 follows dim(A − B)(X) = 1. It means that there exist a vector w ∈ SY and a functional y ∗ ∈ X ∗ such that the operator A − B has the form: (A − B)(x) = y ∗ (x)w,
∀ x ∈ X.
Since dim X = 2, the functional y ∗ attains its norm, hence MA−B = ∅. We show that MA−B consists of two elements only. Assume the converse, i.e., suppose that there exist two linearly independent vectors u1 , u2 ∈ SX such that A − B = Au1 − Bu1 = y ∗ (u1 )w = y ∗ (u1 ) = y ∗ and A − B = Au2 − Bu2 = y ∗ (u2 )w = y ∗ (u2 ) = y ∗ . 2 2 Then y ∗ (u1 ) = y ∗ (u2 ), whence also y ∗ u1 +u = y ∗ . It means that u1 +u ∈ SX and consequently, since 2 2 X is strictly convex, u1 = u2 , a contradiction. Summing up, we have that MA−B = {u1 , −u1 } for some u1 ∈ SX and the space Y is smooth at the point (A − B)(u1 ). Applying Theorem 3.2 we get that A − B is a point of smoothness of the space L(X, Y ). 2 Now, we are going to prove that under condition (3.4), the approximate symmetry of the orthogonality relation yields some restriction in the length of segments lying on the unit sphere. Theorem 3.4. Let X be a real normed space satisfying (3.4) and let ε ∈ (0, 1). If the orthogonality relation ⊥B in X is ε-symmetric, then R(X) ≤ 2ε. Proof. If X is strictly convex, then R(X) = 0 and the assertion follows trivially. Assume now that X is not strictly convex and let conv {x, y} ⊂ SX , for some x, y ∈ SX , x = y. Then x⊥B x − y and y⊥B x − y. Homogeneity of ⊥B yields −y⊥B x − y. From the assumed ε-symmetry of ⊥B it follows that x − y⊥εB x and x − y⊥εB − y. Implication (3.3) and smoothness at x − y implies x − y⊥ερ+ x and x − y⊥ερ+ − y, and finally ρ+ (x − y, x) ≤ εx − y and ρ+ (x − y, −y) ≤ εx − y. According to (3.2), ρ+ is additive with respect to the second variable, hence adding both above inequalities we get ρ+ (x − y, x − y) ≤ 2εx − y. Therefore x − y ≤ 2ε and R(X) ≤ 2ε follows. 2 Example 3.5. The assumption (3.4) is essential. Consider X = R2 with the l∞ -l1 -norm (cf. Example 2.5). Since it is a Radon norm, the Birkhoff orthogonality is symmetric, whence ⊥B is ε-symmetric with an arbitrarily small ε > 0. On the other hand R(X) = 1 and the estimation R(X) ≤ 2ε does not hold.
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
632
If R(X) = 2, then inequality R(X) ≤ 2ε implies ε ≥ 1, which means that the Birkhoff orthogonality is not approximately symmetric. Namely, we have: Corollary 3.6. If X is a real normed space satisfying (3.4) and R(X) = 2, then for all ε ∈ [0, 1) the Birkhoff orthogonality in X is not ε-symmetric. If X is a finite-dimensional space with R(X) = 2, then X cannot be smooth but it may satisfy (3.4). That is why one can apply the above corollary, e.g., for R2 with the maximum norm. For infinite-dimensional spaces it is possible that X is smooth and R(X) = 2. Example 3.7. For a given a ∈ (0, 2) let Xa be a two-dimensional, smooth normed space for which R(X) = a (e.g., a normed space generated by a suitably smoothly squeezed Euclidean ball). (See Fig. 3.)
Fig. 3. Smooth normed space with R(X) = a.
For an := 2 −
1 n
consider a sequence (Xan ) of such spaces and their infinite l2 -product X := Xa1 ⊕2 Xa2 ⊕2 Xa3 ⊕2 . . . .
Then X is an example of a smooth normed space for which R(X) = 2 and it follows from Corollary 3.6 that the Birkhoff orthogonality in X is not approximately symmetric. Symmetry of ⊥B and Theorem 3.4 yield R(X) ≤ 2ε with an arbitrary ε > 0. Hence R(X) = 0, i.e., X is strictly convex. That means, in particular, that a smooth Radon norm is strictly convex (actually, for Radon plane, smoothness and strict convexity are equivalent — cf. [6, Corollary 3.2]). We remind that the modulus of convexity of a normed space X is a mapping δX : [0, 2] → [0, 1] given by x + y : x, y ∈ SX , x − y = ε . δX (ε) := inf 1 − 2 The mapping δX is non-decreasing and continuous in [0, 2). Moreover, one can easily derive x+y 2 ≤ 1 − δX (x − y),
∀ x, y ∈ SX .
(3.5)
Theorem 3.8. If ε ≥ 1 − 2δX (1), then the relation ⊥B is ε-symmetric. Proof. Assume the converse, i.e., that ⊥B is not ε-symmetric. Then there exist two vectors x, y ∈ SX such ϕ(x) that x⊥B y but not y⊥εB x. Fix arbitrarily ϕ ∈ J(y). It follows from (1.3) that |ϕ(x)| > ε. Let λ := |ϕ(x)| . Since x⊥B y, we have in particular 1 = x ≤ x − λy and monotonicity of δX gives δX (1) ≤ δX (x − λy). Thus, by (3.5),
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
633
ϕ x + λy ≤ x + λy ≤ 1 − δX (x − λy) ≤ 1 − δX (1).
2 2 On the other hand
ϕ x + λy = 1 + |ϕ(x)| > 1 + ε
2 2 2 and therefore ε < 1 − 2δX (1), a contradiction to our assumption. 2 Corollary 3.9. Suppose that for any ε ∈ [0, 1) the relation ⊥B is not ε-symmetric. Then ε0 (X) := sup{ε ∈ [0, 2] : δX (ε) = 0} ≥ 1. Moreover, if X is finite-dimensional, then R(X) ≥ 1. Proof. If follows from the assumption that 1 − 2δX (1) ≥ 1, hence δX (1) = 0 and therefore ε0 (X) ≥ 1. If X is finite-dimensional, then ε0 (X) = R(X). 2 4. Approximate symmetry in uniformly convex spaces and their dual spaces If δX (1) > 0 (in particular if X is uniformly convex), then 1 − 2δX (1) < 1. Therefore, it follows from Theorem 3.8 that for any ε ∈ [1 − 2δX (1), 1), the Birkhoff orthogonality is ε-symmetric. Theorem 4.1. Let X be a real uniformly convex normed space. Then, the Birkhoff orthogonality in X is approximately symmetric. If X is a finite-dimensional space, then the uniform convexity of X can be replaced by its smoothness. However, it is not true for infinite-dimensional smooth spaces. Cf. Corollary 3.6 and Example 3.7. Theorem 4.2. Let X be a finite-dimensional real smooth normed space. Then there exists an ε < 1 such that the Birkhoff orthogonality is ε-symmetric. Proof. Assume the converse. Then there exist an increasing sequence εn such that εn 1 and sequences of unit vectors xn , yn such that xn ⊥B yn
and yn ⊥ εBn xn .
The latter condition can be equivalently written as: εn < ϕn (xn ) ,
∀ ϕn ∈ J (yn ) .
We can choose convergent subsequences xnk → x, ynk → y, ϕnk → ϕ (closed balls in X, X ∗ are compact) and get x⊥B y,
ϕ ∈ J(y),
and 1 = ϕ(x).
Since x⊥B y, there exists x∗ ∈ J(x) such that x∗ (y) = 0. Observe that also ϕ(x) = 1, whence ϕ ∈ J(x). Since X is smooth, ϕ = x∗ . Thus 0 = x∗ (y) = ϕ(y) = 1, a contradiction. 2
634
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
In the sequel we consider an approximate orthogonality in various spaces (X, X ∗ and X ∗∗ ). However, we use one common symbol ⊥εB . For a real normed space X we denote by π : X → X ∗∗ the canonical isometric embedding of X into X ∗∗ : π(x) : X ∗ x∗ → x∗ (x) ∈ R. Since π is a linear isometry from X onto π(X), for arbitrary x, y ∈ X and ε ∈ [0, 1) we have x⊥εB y ⇐⇒ π(x)⊥εB π(y).
(4.1)
Theorem 4.3. Let X be a real uniformly convex Banach space. Then, there exists ε ∈ [0, 1) (the same as in Theorem 4.1) such that the Birkhoff orthogonality in X ∗∗ is ε-symmetric. Proof. Uniform convexity in a Banach space implies reflexivity, thus for arbitrary x∗∗ , y ∗∗ ∈ X ∗∗ we have x∗∗ = π(x) and y ∗∗ = π(y) for some x, y ∈ X. Applying (4.1) and Theorem 4.1 we get x∗∗ ⊥B y ∗∗ ⇔ π(x)⊥B π(y) ⇔ x⊥B y ⇒ y⊥εB x ⇔ π(y)⊥εB π(x) ⇔ y ∗∗ ⊥εB x∗∗ , which completes the proof. 2 Suppose now that X is reflexive, smooth and strictly convex. In particular, smoothness guarantees that for each x ∈ X there is a unique supporting functional, denoted by ϕx (that is J(x) = {ϕx }). Moreover, the mapping SX x → ϕx ∈ SX ∗ is injective. Indeed, if ϕx = ϕy , then ϕx (x + y) = ϕx (x) + ϕx (y) = ϕx (x) + ϕy (y) = 1 + 1 = 2. Thus 2 = ϕx (x + y) ≤ x + y ≤ 2, whence x + y = 2 and strict convexity yields x = y. Lemma 4.4. Let X be a real reflexive, strictly convex and smooth Banach space. Then, for any ε ∈ [0, 1) x⊥εB y ⇐⇒ ϕy ⊥εB ϕx ,
∀ x, y ∈ SX .
(4.2)
Proof. Let x, y ∈ SX . It follows then from (1.3) x⊥εB y ⇐⇒ |ϕx (y)| ≤ ε.
(4.3)
On the other hand, we have from (1.3) applied to X ∗ : ϕy ⊥εB ϕx ⇐⇒ ∃ Φ ∈ X ∗∗ s.t. Φ = 1, Φ(ϕy ) = 1, and |Φ(ϕx )| ≤ ε. Reflexivity of X yields that for some a ∈ X, Φ = π(a). Moreover, a = π(a) = Φ = 1 and ϕy (a) = π(a)(ϕy ) = Φ(ϕy ) = 1, which means that ϕy ∈ J(a) = {ϕa }. It follows that ϕy = ϕa , hence a = y. Moreover, |ϕx (y)| = |ϕx (a)| = |π(a)(ϕx )| = |Φ(ϕx )| ≤ ε. Thus we have ϕy ⊥εB ϕx ⇐⇒ |ϕx (y)| ≤ ε. Now (4.2) follows from (4.3) and (4.4).
(4.4)
2
Theorem 4.5. Let X be a real, reflexive, strictly convex and smooth Banach space. If the orthogonality relation ⊥B in X is ε-symmetric (with some ε ∈ [0, 1)), then the orthogonality relation in X ∗ is also ε-symmetric (with the same ε).
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
635
Proof. Assume ε-symmetry of ⊥B in X. Let x∗ , y ∗ ∈ X ∗ and x∗ ⊥B y ∗ . Since the relations ⊥B and ⊥εB are homogeneous, without loss of generality we assume that x∗ , y ∗ ∈ SX ∗ . Reflexivity of X yields that x∗ , y ∗ attain their norms, i.e., there exist x, y ∈ SX such that x∗ ∈ J(x), y ∗ ∈ J(y). Since X is smooth, x∗ = ϕx and y ∗ = ϕy and applying Lemma 4.4 twice and ε-symmetry of ⊥B in X, we get x∗ ⊥B y ∗ ⇔ ϕx ⊥B ϕy ⇔ y⊥B x ⇒ x⊥εB y ⇔ ϕy ⊥εB ϕx ⇔ y ∗ ⊥εB x∗ , which completes the proof. 2 Finally, applying Theorems 4.1, 4.5 and 4.3, we conclude. Theorem 4.6. Let X be a real uniformly convex and smooth Banach space. Then, there exists ε ∈ [0, 1) such that the Birkhoff orthogonality relations in X, X ∗ and X ∗∗ are ε-symmetric (with the same ε). 5. Birkhoff orthogonality symmetry constant Let X be a normed space with dim X ≥ 2. For X we define a constant connected with the symmetry of the Birkhoff orthogonality: S(X) := inf{ε ∈ [0, 1] : ∀ x, y ∈ X x⊥B y ⇒ y⊥εB x}. Remark 2.2 yields that, in fact, the infimum in the above definition can be replaced by minimum. Obviously, S(X) ∈ [0, 1] and S(X) = 0 means that ⊥B is symmetric, i.e., X is a Radon plane or an inner product space. S(X) = 1 means that ⊥B is not ε-symmetric with any ε ∈ [0, 1), i.e., ⊥B is not approximately symmetric. Bearing in mind the results obtained in previous sections of the paper, we can immediately formulate some further properties of S(X). • From Theorem 3.8 we get for an arbitrary real normed space X: S(X) ≤ 1 − 2δX (1) ≤ 1.
(5.1)
It follows that if δX (1) > 0 (in particular if X is uniformly convex), then S(X) < 1. The reverse is not true. For, let consider a finite-dimensional smooth space X with R(X) ≥ 1 (cf. Example 3.7 and the space Xa with a > 1). Then δX (1) = 0 whereas it follows from Theorem 4.2 that S(X) < 1. • If X satisfies (3.4), Theorem 3.4 yields 1 R(X) ≤ S(X). 2
(5.2)
The assumption (3.4) is essential, cf. Example 3.5. • If X is a real uniformly convex and smooth Banach space, then applying Theorem 4.6 we get S(X) = S(X ∗ ) = S(X ∗∗ ) < 1.
(5.3)
Proposition 5.1. Let X be a normed space. Then S(X) = sup{S(X0 ) : X0 is a two-dimensional subspace of X}.
(5.4)
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
636
Proof. Denote S2 (X) := sup{S(X0 ) : X0 is a two-dimensional subspace of X}. We will show that S2 (X) = S(X). Since X0 ⊂ X, S(X0 ) ≤ S(X), whence S2 (X) ≤ S(X). If S2 (X) = 1, then S(X) = 1. Assume S2 (X) < 1. For n ∈ N such that S2 (X) + n1 ≤ 1 let ε ∈ S2 (X), S2 (X) + n1 . Let x, y ∈ X, x⊥B y, X0 := Lin{x, y}. There is S(X0 ) ≤ S2 (X) < ε < S2 (X) + n1 . Since ε > S(X0 ) and x, y ∈ X0 , we have y⊥εB x. Thus we have proved x⊥B y ⇒ y⊥εB x whence S(X) ≤ ε < S2 (X) + n1 . Letting n → ∞, S(X) ≤ S2 (X) and finally S(X) = S2 (X). 2 Example 5.2. Let X = R3 with the norm (x, y, z) = max{ x2 + y 2 , |z|},
∀ (x, y, z) ∈ R3 .
Let X1 = {(x, y, 0) : x, y ∈ R}
and X2 = {(0, y, z) : y, z ∈ R}.
Then S(X1 ) = 0, S(X2 ) = 1 and S(X) = 1. Under some additional assumptions we will improve the estimation (5.1). Theorem 5.3. Let (X, · ) be a real, uniformly smooth Banach space. Suppose that · 2 is an equivalent norm in X: ax ≤ x2 ≤ bx,
∀x ∈ X
(5.5)
2 (with some 0 < a ≤ b) such that H := (X, · 2 ) is a Hilbert space. Finally, assume that 1 − 2δX ∗ (1) < ab . Then
2 1 − ab −1 . (5.6) S(X) ≤ δX ∗ 2 Proof. Fix x, y ∈ X and assume x⊥B y. Then for any λ ∈ R we have a x2 ≤ ax ≤ ax + λy ≤ x + λy2 b and it follows that x⊥ B y in H (cf. (1.5)) with δ := 1 − ab . Since H is a Hilbert space, the approximate δ orthogonality is symmetric whence we have also y⊥ B x. It gives δ
a y2 ≤ y + λx2 , b and together with (5.5) a 1 a2 y ≤ 2 y2 ≤ y + λx2 ≤ y + λx. 2 b b b Defining γ := 1 −
a 2
we have (1 − γ)y ≤ y + λx, i.e., y⊥ B x in X. Applying Theorem 1.4 (notice that γ −1 γ γ < 2δX ∗ (1)) we obtain y⊥ηB x with η = δX . ∗ 2 Since x, y were chosen arbitrarily, we have proved that the orthogonality relation ⊥B in X is η-symmetric. Thus, in particular, S(X) ≤ η. 2 b
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
637
Let d(X, Y ) denote the Banach–Mazur distance of isomorphic spaces X and Y , i.e., d(X, Y ) := inf{T T −1 : T : X → Y is a linear isomorphism onto Y } ≥ 1. If the estimation (5.5) is sharp, then d(H, X) =
b a
and Theorem 5.3 can be reformulated as follows.
Theorem 5.4. Let H be a Hilbert space and let X be a uniformly smooth Banach space. Assume d(H, X) ≤
1 1 − 2δX ∗ (1)
.
Then
S(X) ≤
−1 δX ∗
1−
1 d(H,X)2
2
.
If a Banach space X is uniformly smooth, then it is reflexive and X ∗ is uniformly convex. Thus −1 δX ∗ : [0, 2] → [0, 1] is a continuous increasing bijection and hence δX ∗ : [0, 1] → [0, 2] is also continuous, increasing and bijective. Our previous theorem yields that if d(H, X) → 1+ (for some Hilbert space H), then S(X) → 0 = S(H). It means some kind of continuity of X → S(X). However, the following example shows that in general this correspondence is not continuous. Example 5.5. Let X = R2 with an l1 -l∞ norm, i.e., with the unit sphere being a hexagon with vertices at x1 = (1, 0), x2 = (0, 1), x3 = (−1, 1), x4 = (−1, 0), x5 = (0, −1), x6 = (1, −1). For an arbitrary positive constant δ let Y = Yδ = R2 with the norm such that the unit sphere is a hexagon with vertices at y1 = (1, 0), y2 = (0, 1), y3 = (−1 − δ, 1 − δ), y4 = (−1, 0), y5 = (0, −1), y6 = (1 + δ, −1 + δ). See Fig. 4.
Fig. 4. Example 5.5, unit spheres in X and Y .
Let T : X → Y be a linear mapping defined by T x2 = y2 and T x3 = y3 , whence T (x, y) = ((1 + δ)x, δx + y),
∀ (x, y) ∈ R2
and T
−1
(x, y) =
δ 1 x, y − x , 1+δ 1+δ
∀ (x, y) ∈ R2 .
638
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
1+δ 1+δ One can notice that T (BX ) ⊂ 1−δ BY and T −1 (BY ) ⊂ BX . Therefore T ≤ 1−δ and T −1 ≤ 1 and −1 hence T T → 1 as δ → 0. It means that, for an arbitrary η > 0 and with sufficiently small δ, we have d(X, Y ) < 1 + η. The space X is a Radon plane, therefore S(X) = 0. For the space Y we can see that R(Y ) > 1. Moreover, the condition (3.4) is satisfied. Thus it follows from (5.2) that
1 1 < R(Y ) ≤ S(Y ). 2 2 To conclude, for the space X we have S(X) = 0 and for an arbitrary η > 0 there exists a normed space Y such that d(X, Y ) < 1 + η, and S(Y ) > 12 . In the following example we give some estimation for S(lnp ). Example 5.6. Suppose that p, q > 1 are conjugated (i.e., p1 + 1q = 1). Consider the space lnp and its dual lnq . It is known (cf. [11, Lemma 2]) that the Banach–Mazur distance between lnp and ln2 is equal to: d(lnp , ln2 )
d :=
1 1
p−2
=n
(consequently d(lnq , ln2 ) = d(lnp , ln2 ) = d). Suppose that p and q are sufficiently close to 2, namely such that
1 1
p−2
n
1 1
q−2
=n
≤ min
1 1 − 2δlp (1)
,
1 1 − 2δlq (1)
Then it follows from Theorem 5.4 S(lnp )
≤
δl−1 q n
1 − d12 2
,
and by monotonicity of δ we have δlnq (S(lnp )) ≤
1 − d12 . 2
Assume now 1 < p ≤ 2. Then q ≥ 2 and we have (cf. [12, Theorem 2]) ε q q1 δlnq (ε) = 1 − 1 − 2 whence we get 1 1 − d12 S(lnp )q q p q (S(l )) ≤ . 1− 1− = δ l n n 2q 2 Thus, for p ≤ 2 and such that (5.7) holds, we have S(lnp )
≤
1 2 − 1+ 2 d q
q q1 .
.
(5.7)
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
639
If p ≥ 2, then q ≤ 2 and interchanging the roles of p and q we get S(lnq )
p p1 1 p ≤ 2 − 1+ 2 . d
But S(lnq ) = S(lnp ) (see (5.3)), so we get S(lnp )
2q − 1 + ≤ 2p − 1 +
1 1 q q d2 1 1 p p d2
,
1 < p ≤ 2,
, p ≥ 2.
It follows that for any p > 1 satisfying (5.7) we have S(lnp )
≤ max
1 2 − 1+ 2 d q
q q1 p p1 1 p , 2 − 1+ 2 . d
Notice that if p → 2, then q → 2 and d → 1. Therefore the right hand side of the above estimation tends to zero, which means limp→2 S(lnp ) = S(ln2 ). Let X be a real Banach space and let Y be a closed subspace of X. A continuous linear operator P : X → Y is called a projection onto Y if P |Y = Id|Y . By P(X, Y ) we denote the family of all projections from X to Y . If P(X, Y ) = ∅, we define a relative projection constant of Y with respect to X as λ(Y, X) := inf{P : P ∈ P(X, Y )}. If there exists a projection P : X → Y satisfying P = λ(Y, X), we call P a minimal projection. In particular, if dim X < ∞, then any subspace Y admits a (not necessarily unique) minimal projection. Cf. [20] for basic information on this topic. The case λ(Y, X) = 1 attracts much attention (cf. a survey [21]); Y is called then a one-complemented subspace of X. Another problem is to give a quantitative estimation of the relative projection constant. The classical result of Bohnenblust is quoted below. Recent results, as well as more information on the problem, can be found, e.g., in [17,18]. Theorem 5.7 ([4]). Let X be an n-dimensional normed space and let Y be a subspace of X with dim Y = n−1. Then, λ(Y, X) ≤ 2 − n2 . If the Birkhoff orthogonality is ε-symmetric, the above estimation can be improved. Theorem 5.8. Let X be a real reflexive space for which ⊥B is ε-symmetric with some ε ∈ 0, 12 . Let Y be a 1 closed subspace with codim Y = 1. Then λ(Y, X) ≤ 1−2ε . Proof. Assuming ε-symmetry of ⊥B and applying (1.6), we get x⊥B y
⇒
y⊥ B x. 2ε
(5.8)
Let Y be a closed subspace of X with codim Y = 1. Since X is reflexive, there exists x ∈ SX such that x⊥B Y (see [16] or [1, Theorem 4.13]). Applying (5.8), we get Y ⊥ B x. 2ε
(5.9)
640
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
For each vector y + αx ∈ X = Y + Lin{x} (unique decomposition) we define P (y + αx) := y. Then P ∈ P(X, Y ) and, by (5.9), (1 − 2ε)P (y + αx) = (1 − 2ε)y ≤ y + αx. Thus for each y + αx ∈ X we have P (y + αx) ≤ 1 we get λ(Y, X) ≤ 1−2ε . 2
1 1−2ε y + αx,
whence P ≤
1 1−2ε .
Since λ(Y, X) ≤ P ,
If X is an n-dimensional normed space and the orthogonality relation ⊥B is ε-symmetric with ε < 12 − 4−1 4 , n then the estimation of λ(Y, X) obtained in Theorem 5.8 is better than in Theorem 5.7. Moreover, the estimation in Theorem 5.8 does not depend on n. Finally, as opposed to Theorem 5.7, Theorem 5.8 is valid without any restriction on the dimension of the normed space X. As a corollary we get another estimation of the symmetry constant S(X). Theorem 5.9. Let X be a real reflexive normed space and let Y be a closed subspace of X with codim Y = 1. Then 1 1 1− . S(X) ≥ 2 λ(Y, X) References [1] J. Alonso, H. Martini, S. Wu, On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces, Aequationes Math. 83 (2012) 153–189. [2] C. Alsina, J. Sikorska, M.S. Tomás, Norm Derivatives and Characterizations of Inner Product Spaces, World Scientific, Hackensack, NJ, 2010. [3] G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J. 1 (1935) 169–172. [4] F. Bohnenblust, Convex regions and projections in Minkowski spaces, Ann. of Math. 39 (1938) 301–308. [5] J. Chmieliński, On an ε-Birkhoff orthogonality, J. Inequal. Pure Appl. Math. 6 (3) (2005) 79. [6] J. Chmieliński, Operators reversing orthogonality in normed spaces, Adv. Oper. Theory 1 (1) (2016) 8–14. [7] J. Chmieliński, P. Wójcik, ρ-orthogonality and its preservation-revisited, in: Recent Developments in Functional Equations and Inequalities, in: Banach Center Publications, vol. 99, 2013, pp. 17–30. [8] J. Chmieliński, T. Stypuła, P. Wójcik, Approximate orthogonality in normed spaces and its applications, Linear Algebra Appl. 531 (2017) 305–317. [9] S.S. Dragomir, On approximation of continuous linear functionals in normed linear spaces, An. Univ. Timiş. Ser. Ştiinţ. Mat. 29 (1991) 51–58. [10] P.M. Gruber, Stability of Blaschke’s characterization of ellipsoids and Radon norms, Discrete Comput. Geom. 17 (1997) 411–427. [11] V.E. Gurarii, M.I. Kadec, V.E. Macaev, On the distance between isomorphic spaces of finite dimension, Math. Sb. 70 (1966) 481–489 (in Russian). [12] O. Hanner, On the uniform convexity of Lp and lp , Ark. Mat. 3 (1956) 239–244. [13] S. Heinrich, The differentiability of the norm in spaces of operators, Funct. Anal. Appl. 9 (1976) 360–362; transl. from Funktsional. Anal. i Prilozhen. 9 (4) (1975) 93–94. [14] R.C. James, Orthogonality in normed linear spaces, Duke Math. J. 12 (1945) 291–301. [15] R.C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947) 265–292. [16] R.C. James, Reflexivity and the sup of linear functionals, Israel J. Math. 13 (1972) 289–300. [17] T. Kobos, Hyperplanes of finite-dimensional normed spaces with the maximal relative projection constant, Bull. Aust. Math. Soc. 91 (2015) 447–463. [18] T. Kobos, A uniform estimate of the relative projection constant, J. Approx. Theory 225 (2018) 58–75. [19] B. Mojškerc, A. Turnšek, Mappings approximately preserving orthogonality in normed spaces, Nonlinear Anal. 73 (2010) 3821–3831. [20] W. Odyniec, G. Lewicki, Minimal Projections in Banach Spaces, Lecture Notes in Mathematics, vol. 1449, Springer, Berlin, Heidelberg, New York, 1990. [21] B. Randrianantoanina, Norm-one projections in Banach spaces, Taiwanese J. Math. 5 (2001) 35–95. [22] T. Stypuła, P. Wójcik, Characterizations of rotundity and smoothness by approximate orthogonalities, Ann. Math. Sil. 30 (2016) 183–201. [23] P. Wójcik, Norm-parallelism in classical M-ideals, Indag. Math. 28 (2017) 287–293.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Approximate symmetry of Birkhoff orthogonality Jacek Chmieliński ∗ , Paweł Wójcik Department of Mathematics, Pedagogical University of Cracow, Podchor¸ ażych 2, 30-084 Kraków, Poland
a r t i c l e
i n f o
Article history: Received 23 November 2017 Available online 31 January 2018 Submitted by T. Domínguez Benavides
a b s t r a c t In a real normed space X we consider an approximate symmetry of the Birkhoff orthogonality ⊥B and establish its connections with some properties of the space X. Moreover, we introduce and study a new geometric constant for X, connected with the considered property. © 2018 Elsevier Inc. All rights reserved.
Keywords: Birkhoff orthogonality Approximate Birkhoff orthogonality Approximate symmetry of Birkhoff orthogonality
0. Introduction The Birkhoff orthogonality in a normed space is generally not a symmetric relation. However, allowing some inaccuracy one can consider an approximate Birkhoff orthogonality (a new characterization of it has been obtained recently in [8]) and then a related notion of approximate symmetry of this orthogonality can be introduced and investigated. In the present paper we deal with this problem. We consider also some geometrical properties connected with such an approximate symmetry. Throughout the paper we consider a real normed space (X, · ) with dim X ≥ 2. BX and SX stand for the closed unit ball and the unit sphere in X, respectively. For another normed space Y , L(X, Y ) denotes the space of all linear and bounded operators from X into Y . For T ∈ L(X, Y ), by MT we mean the set of unit vectors at which T attains its norm, namely MT := {x ∈ SX : T x = T }. K(X, Y ) is the subspace of L(X, Y ) consisting of all compact operators. X ∗ stands for the dual and X ∗∗ for the bidual space of X. 1. Approximate Birkhoff orthogonality The Birkhoff orthogonality in X is defined by: x⊥B y
⇐⇒
x + λy ≥ x, ∀ λ ∈ R
* Corresponding author. E-mail addresses: [email protected] (J. Chmieliński), [email protected] (P. Wójcik). https://doi.org/10.1016/j.jmaa.2018.01.031 0022-247X/© 2018 Elsevier Inc. All rights reserved.
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
626
(cf. [3,14,15] or a survey [1]). We consider also an approximate Birkhoff orthogonality (ε-Birkhoff orthogonality with ε ∈ [0, 1)): ⇐⇒
x⊥εB y
x + λy2 ≥ x2 − 2εx λy, ∀ λ ∈ R
(1.1)
as introduced in [5]. Obviously, ⊥0B = ⊥B . If X is an inner product space, the approximate orthogonality is naturally defined by: ⇐⇒
x⊥ε y
| x|y | ≤ ε x y,
and in this case ⊥εB coincides with ⊥ε (cf. [5, Proposition 2.1]). In a recent paper [8] the authors have proved the following characterization of the approximate Birkhoff orthogonality. Theorem 1.1 ([8], Theorem 2.3). Let X be a real normed space. For x, y ∈ X and ε ∈ [0, 1): ⇐⇒
x⊥εB y
∃z ∈ Lin{x, y} s.t. x⊥B z, z − y ≤ εy.
(1.2)
Another characterization can be derived from Theorem 1.1 by using the supporting functionals at x ∈ X: J(x) := {ϕ ∈ X ∗ :
ϕ = 1, ϕ(x) = x}.
Theorem 1.2 ([8], Theorem 2.4). Let X be a real normed space. For x, y ∈ X and ε ∈ [0, 1): ⇐⇒
x⊥εB y
∃ ϕ ∈ J(x) s.t. |ϕ(y)| ≤ εy.
(1.3)
Clearly, (1.3) generalizes James’ characterization (cf. [15, Corollary 2.2]): x⊥B y
⇐⇒
∃ϕ ∈ J(x) s.t. ϕ(y) = 0.
(1.4)
A different definition of an approximate Birkhoff orthogonality was given by Dragomir [9]. For a given ε ∈ [0, 1) and x, y ∈ X: x⊥ B y ε
⇐⇒
x + λy ≥ (1 − ε)x, ∀ λ ∈ R.
(1.5)
Some relationships between definitions (1.1) and (1.5) were established in [5] and [19]. In particular, for η inner product spaces ⊥ B is equal to ⊥ with η = 1 − (1 − ε)2 . ε
In an arbitrary normed space Proposition 3.1 in [19] yields x⊥εB y with δ := 1 −
√
⇒
x⊥ B y, δ
1 − 4ε and for ε ≤ 14 . We will improve this result for real spaces.
Theorem 1.3. Let X be a real normed space and let ε ∈ 0, 12 . Then x⊥εB y
⇒
x⊥ B y. 2ε
(1.6)
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
627
Proof. Assume that for x, y ∈ X \ {0} and ε ∈ 0, 12 , we have x⊥εB y. Let z ∈ Lin{x, y} be as in (1.2). The mapping ϕ : R → R, ϕ(t) := x − ty is nonnegative, convex and limt→±∞ ϕ(t) = +∞. Thus it attains its minimum, i.e., there exists β ∈ R such that min{x − αy : α ∈ R} = x − βy. Therefore, for any α ∈ R we get |β| y − x ≤ βy − x ≤ x − αy. Taking α = 0 we get |β| ≤ Now, for an arbitrary α ∈ R we have (1 − 2ε)x = x − 2εx = x −
(1.7) 2x . y
2x εy ≤ x − |β| z − y y
= x − βz − βy ≤ x − βz − βz − βy ≤ (x − βz) + (βz − βy) = x − βy ≤ x − αy, i.e., x⊥ B y. 2ε
2
Yet another connection between the two considered relations was also established in [19]. We will use this result in Section 5. Here δX ∗ denotes the modulus of convexity for the dual space X ∗ . Theorem 1.4 ([19], Proposition 3.8). Let X be a real uniformly smooth normed space and let ε ∈ [0, 2δX ∗ (1)). Then x, y ∈ X, x⊥ B y ε
−1 where η = δX ∗
=⇒
x⊥ηB y,
ε 2 .
2. Approximate symmetry of the Birkhoff orthogonality The Birkhoff orthogonality is, generally, not symmetric (Fig. 1 describes R2 with the maximum norm and vectors x, y such that x⊥B y but y ⊥ B x).
Fig. 1. R2 with the maximum norm; x⊥B y, y ⊥ B x.
If X is a normed space with dim X ≥ 3, then ⊥B is symmetric if and only if X is an inner product space. For 2-dimensional linear spaces it is possible, however, to find a norm (called a Radon norm, cf. [1]) which does not come from an inner product but its Birkhoff orthogonality is symmetric. One can relax a little bit the symmetry requirement and consider an approximate symmetry.
628
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
Definition 2.1. The Birkhoff orthogonality relation in a normed space X is called approximately symmetric, or more precisely: ε-symmetric for some ε ∈ [0, 1), if for any x, y ∈ X: x⊥B y
y⊥εB x.
=⇒
(2.1)
Remark 2.2. If the Birkhoff orthogonality relation is ε-symmetric, then it is also ε -symmetric with an arbitrary ε > ε. On the other hand, if the relation ⊥B is εn -symmetric with εn ∈ [0, 1), n = 1, 2, . . . and εn ε0 , then ⊥B is also ε0 -symmetric. The problem of finding the smallest ε for which the approximate symmetry holds true will be considered in Section 5. Theorem 1.1 yields an immediate characterization of ε-symmetry. Theorem 2.3. The Birkhoff orthogonality in a real normed space X is ε-symmetric for some ε ∈ [0, 1) if and only if for x, y ∈ X: x⊥B y
=⇒
∃ z ∈ Lin{x, y} :
y⊥B z, z − x ≤ εx.
(2.2)
Remark 2.4. Gruber [10] introduced a similar definition: the Birkhoff orthogonality is symmetric up to ε if and only if, for all u, v ∈ SX : u⊥B v
=⇒
∃ w ∈ SX :
v⊥B w, w − u ≤ ε.
(2.3)
It is easy to see that the above definition is equivalent to: x⊥B y
=⇒
∃z ∈ X :
z = x, y⊥B z, z − x ≤ εx
(2.4)
for all x, y ∈ X. Thus the notions of ε-symmetry and of Gruber’s symmetry up to ε practically coincide. Indeed, it is clear that if ⊥B is symmetric up to ε, then it is also ε-symmetric. Conversely, one can show 2ε that if ⊥B is ε-symmetric, then (after replacing z by z := xz z ) it is also symmetric up to ε := 1−ε (hence it is necessary to assume ε < 1/3 in order to keep ε < 1). As we will see, the approximate symmetry of the Birkhoff orthogonality does not imply that the norm comes from an inner product (even if dim X ≥ 3). On the other hand, there are spaces in which the orthogonality relation is not approximately symmetric, i.e., for any ε ∈ [0, 1), ⊥B is not ε-symmetric. Indeed, consider X = R2 with the maximum norm. For x = (1, 1) and y = (0, 1) we have x⊥B y. Suppose that y⊥εB x holds for some ε ∈ [0, 1). It would mean that y⊥B z and z − x ≤ ε for some z. Then z = (α, 0) for some α ∈ R and z − x ≥ 1, a contradiction. See also Corollary 3.6 and Example 3.7. Bearing in mind definition (1.5) and property (1.6), we notice that for ε ∈ 0, 12 condition (2.1) implies x⊥B y
=⇒
y⊥ B x. 2ε
(2.5)
In some cases the above condition may be more convenient for applications and in particular we will use it in Section 5. Symmetry of the approximate orthogonality yields approximate symmetry of the orthogonality. Namely, for an arbitrary normed space X and ε ∈ [0, 1): ⊥εB is symmetric
=⇒
⊥B is ε-symmetric.
(2.6)
Indeed, if x⊥B y, then also x⊥εB y (with an arbitrary ε ∈ [0, 1)) and the assumed symmetry of ⊥εB yields y⊥εB x.
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
629
Let εn ∈ (0, 1), n = 1, 2, . . . and εn → 0 with n → ∞. Then we have: ⊥εBn is symmetric for all n ∈ N
=⇒
⊥B is symmetric.
(2.7)
Indeed, assume x⊥B y; then, for n ∈ N, also x⊥εBn y and it follows from the assumption that y⊥εBn x. Hence there exist zn ∈ Lin{x, y} such that x − zn ≤ εn x and y⊥B zn (n ∈ N). Letting n → ∞, we get zn → x and y⊥B x. On the other hand, symmetry of the Birkhoff orthogonality does not imply symmetry of the approximate orthogonality. Example 2.5. Let X = R2 with the l∞ -l1 -norm: max{|x|, |y|}, (x, y) = |x| + |y|,
if xy ≥ 0; if xy < 0.
The above norm is a Radon norm, i.e., the Birkhoff orthogonality relation is symmetric. (See Fig. 2.)
Fig. 2. R2 with (a Radon) l∞ -l1 -norm; x⊥εB y, y ⊥εB x.
Let x = (1, 0), y = (ε, ε − 1) with some arbitrarily fixed ε ∈ (0, 1). For z = (0, −1) we have x⊥B z and y − z = (ε, ε) = ε = εz, whence x⊥εB y. Suppose that y⊥εB x. Then it would exist w such that y⊥B w and w − x ≤ εx = ε. If y⊥B w, then w must be of the form w = (t, t) with some t ∈ R. Then w − x = (t − 1, t) ≥ 1 > ε, — a contradiction. Hence y⊥εB x. 3. Geometrical properties related to the approximate symmetry We recall the notion of norm derivatives (cf. [2]): x + ty2 − x2 2t t→0± x + ty − x = x · lim , t t→0±
ρ± (x, y) := lim
∀ x, y ∈ X, t ∈ R.
For each x ∈ X, ρ± (x, x) = x2 . Generally, ρ− ≤ ρ+ . If x0 ∈ X \ {0} is a point of smoothness, then the norm is Gateaux differentiable at x0 (cf. [2, Definition 2.1.1]) whereas ρ− (x0 , y) = ρ+ (x0 , y), In particular, if X is smooth, then ρ− and ρ+ coincide.
∀ y ∈ X.
(3.1)
630
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
The mappings ρ+ and ρ− are subadditive and superadditive, respectively, in the second variable (cf. [2, Proposition 2.1.3]). It follows from (3.1) that for x0 ∈ X \ {0} being a point of smoothness we have ρ± (x0 , y + z) = ρ± (x0 , y) + ρ± (x0 , z),
∀ y, z ∈ X.
(3.2)
In particular, for smooth spaces, both mappings ρ± are additive in the second variable. Norm derivatives yield orthogonality relations: x⊥ρ± y
⇐⇒
ρ± (x, y) = 0,
and related to them approximate orthogonalities: x⊥ερ± y
⇐⇒
|ρ± (x, y)| ≤ εx y.
It is known that ⊥ερ± ⊂ ⊥εB (cf. [7, Theorem 3.2]). The reverse inclusion is, generally, not true unless some kind of smoothness is assumed. Proposition 3.1. If the norm is smooth at x0 ∈ X \ {0}, then x0 ⊥εB y =⇒ x0 ⊥ερ± y,
∀ y ∈ X.
(3.3)
In particular, the equality ⊥ερ± = ⊥εB holds true if X is smooth. Proof. Assuming x0 ⊥εB y, we have (cf. [7, Theorem 3.1]) ρ− (x0 , y) − εx0 y ≤ 0 ≤ ρ+ (x0 , y) + εx0 y. Since X is smooth at x0 , we have ρ− (x0 , ·) = ρ+ (x0 , ·) whence ρ± (x0 , y) − εx0 y ≤ 0 ≤ ρ± (x0 , y) + εx0 y, and hence |ρ± (x0 , y)| ≤ εx0 y, i.e., x0 ⊥ερ± y.
2
For a normed space X we define (cf. [22]) a constant R(X) := sup{x − y : conv {x, y} ⊂ SX }. We will also consider the following property of X which is essentially weaker than smoothness or strict convexity of X: x, y ∈ X, x = y, conv {x, y} ⊂ SX =⇒ X is smooth at x − y.
(3.4)
Obviously, smooth spaces as well as strictly convex ones satisfy (3.4). On the other hand, it is easy to see that R2 with the maximum norm, which is neither strictly convex nor smooth, satisfies (3.4). Actually, one can consider X = R2 with any norm for which the unit ball is a symmetric polygon such that its sides are not parallel to diagonals. Then the condition (3.4) is satisfied. Notice that the property (3.4) is hereditary, i.e., if X satisfies this property then so does any subspace of X. Theorem 3.3 below gives a large family of spaces satisfying (3.4). In the proof of this theorem we will apply a characterization of points of smoothness in the space of compact operators K(X, Y ).
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
631
Theorem 3.2 ([13], Corollary 2.2). Let X, Y be Banach spaces and let X be reflexive. Then a compact operator T is smooth in K(X, Y ) if and only if there exists x0 ∈ SX such that MT = {x0 , −x0 } and T x0 is smooth in Y . Theorem 3.3. Let X be a two-dimensional strictly convex normed space and let Y be a strictly convex and smooth normed space. Then L(X, Y ) (and each its subspace) satisfies the condition (3.4). Proof. Obviously, in our case K(X, Y ) = L(X, Y ). Let A, B ∈ L(X, Y ) such that A = B, A = B = 1 and conv{A, B} ⊂ SL(X,Y ) be fixed. Then A +B = A +B. It follows from [23, Theorem 2.1] that there exists a vector x0 ∈ SX such that Ax0 = A, Bx0 = B, Ax0 + Bx0 = Ax0 + Bx0 . Therefore Ax +Bx 0 0 = 1. Strict convexity of Y yields Ax0 = Bx0 . Thus x0 ∈ ker(A − B) and since dim X = 2, it 2 follows dim(A − B)(X) = 1. It means that there exist a vector w ∈ SY and a functional y ∗ ∈ X ∗ such that the operator A − B has the form: (A − B)(x) = y ∗ (x)w,
∀ x ∈ X.
Since dim X = 2, the functional y ∗ attains its norm, hence MA−B = ∅. We show that MA−B consists of two elements only. Assume the converse, i.e., suppose that there exist two linearly independent vectors u1 , u2 ∈ SX such that A − B = Au1 − Bu1 = y ∗ (u1 )w = y ∗ (u1 ) = y ∗ and A − B = Au2 − Bu2 = y ∗ (u2 )w = y ∗ (u2 ) = y ∗ . 2 2 Then y ∗ (u1 ) = y ∗ (u2 ), whence also y ∗ u1 +u = y ∗ . It means that u1 +u ∈ SX and consequently, since 2 2 X is strictly convex, u1 = u2 , a contradiction. Summing up, we have that MA−B = {u1 , −u1 } for some u1 ∈ SX and the space Y is smooth at the point (A − B)(u1 ). Applying Theorem 3.2 we get that A − B is a point of smoothness of the space L(X, Y ). 2 Now, we are going to prove that under condition (3.4), the approximate symmetry of the orthogonality relation yields some restriction in the length of segments lying on the unit sphere. Theorem 3.4. Let X be a real normed space satisfying (3.4) and let ε ∈ (0, 1). If the orthogonality relation ⊥B in X is ε-symmetric, then R(X) ≤ 2ε. Proof. If X is strictly convex, then R(X) = 0 and the assertion follows trivially. Assume now that X is not strictly convex and let conv {x, y} ⊂ SX , for some x, y ∈ SX , x = y. Then x⊥B x − y and y⊥B x − y. Homogeneity of ⊥B yields −y⊥B x − y. From the assumed ε-symmetry of ⊥B it follows that x − y⊥εB x and x − y⊥εB − y. Implication (3.3) and smoothness at x − y implies x − y⊥ερ+ x and x − y⊥ερ+ − y, and finally ρ+ (x − y, x) ≤ εx − y and ρ+ (x − y, −y) ≤ εx − y. According to (3.2), ρ+ is additive with respect to the second variable, hence adding both above inequalities we get ρ+ (x − y, x − y) ≤ 2εx − y. Therefore x − y ≤ 2ε and R(X) ≤ 2ε follows. 2 Example 3.5. The assumption (3.4) is essential. Consider X = R2 with the l∞ -l1 -norm (cf. Example 2.5). Since it is a Radon norm, the Birkhoff orthogonality is symmetric, whence ⊥B is ε-symmetric with an arbitrarily small ε > 0. On the other hand R(X) = 1 and the estimation R(X) ≤ 2ε does not hold.
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
632
If R(X) = 2, then inequality R(X) ≤ 2ε implies ε ≥ 1, which means that the Birkhoff orthogonality is not approximately symmetric. Namely, we have: Corollary 3.6. If X is a real normed space satisfying (3.4) and R(X) = 2, then for all ε ∈ [0, 1) the Birkhoff orthogonality in X is not ε-symmetric. If X is a finite-dimensional space with R(X) = 2, then X cannot be smooth but it may satisfy (3.4). That is why one can apply the above corollary, e.g., for R2 with the maximum norm. For infinite-dimensional spaces it is possible that X is smooth and R(X) = 2. Example 3.7. For a given a ∈ (0, 2) let Xa be a two-dimensional, smooth normed space for which R(X) = a (e.g., a normed space generated by a suitably smoothly squeezed Euclidean ball). (See Fig. 3.)
Fig. 3. Smooth normed space with R(X) = a.
For an := 2 −
1 n
consider a sequence (Xan ) of such spaces and their infinite l2 -product X := Xa1 ⊕2 Xa2 ⊕2 Xa3 ⊕2 . . . .
Then X is an example of a smooth normed space for which R(X) = 2 and it follows from Corollary 3.6 that the Birkhoff orthogonality in X is not approximately symmetric. Symmetry of ⊥B and Theorem 3.4 yield R(X) ≤ 2ε with an arbitrary ε > 0. Hence R(X) = 0, i.e., X is strictly convex. That means, in particular, that a smooth Radon norm is strictly convex (actually, for Radon plane, smoothness and strict convexity are equivalent — cf. [6, Corollary 3.2]). We remind that the modulus of convexity of a normed space X is a mapping δX : [0, 2] → [0, 1] given by x + y : x, y ∈ SX , x − y = ε . δX (ε) := inf 1 − 2 The mapping δX is non-decreasing and continuous in [0, 2). Moreover, one can easily derive x+y 2 ≤ 1 − δX (x − y),
∀ x, y ∈ SX .
(3.5)
Theorem 3.8. If ε ≥ 1 − 2δX (1), then the relation ⊥B is ε-symmetric. Proof. Assume the converse, i.e., that ⊥B is not ε-symmetric. Then there exist two vectors x, y ∈ SX such ϕ(x) that x⊥B y but not y⊥εB x. Fix arbitrarily ϕ ∈ J(y). It follows from (1.3) that |ϕ(x)| > ε. Let λ := |ϕ(x)| . Since x⊥B y, we have in particular 1 = x ≤ x − λy and monotonicity of δX gives δX (1) ≤ δX (x − λy). Thus, by (3.5),
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
633
ϕ x + λy ≤ x + λy ≤ 1 − δX (x − λy) ≤ 1 − δX (1).
2 2 On the other hand
ϕ x + λy = 1 + |ϕ(x)| > 1 + ε
2 2 2 and therefore ε < 1 − 2δX (1), a contradiction to our assumption. 2 Corollary 3.9. Suppose that for any ε ∈ [0, 1) the relation ⊥B is not ε-symmetric. Then ε0 (X) := sup{ε ∈ [0, 2] : δX (ε) = 0} ≥ 1. Moreover, if X is finite-dimensional, then R(X) ≥ 1. Proof. If follows from the assumption that 1 − 2δX (1) ≥ 1, hence δX (1) = 0 and therefore ε0 (X) ≥ 1. If X is finite-dimensional, then ε0 (X) = R(X). 2 4. Approximate symmetry in uniformly convex spaces and their dual spaces If δX (1) > 0 (in particular if X is uniformly convex), then 1 − 2δX (1) < 1. Therefore, it follows from Theorem 3.8 that for any ε ∈ [1 − 2δX (1), 1), the Birkhoff orthogonality is ε-symmetric. Theorem 4.1. Let X be a real uniformly convex normed space. Then, the Birkhoff orthogonality in X is approximately symmetric. If X is a finite-dimensional space, then the uniform convexity of X can be replaced by its smoothness. However, it is not true for infinite-dimensional smooth spaces. Cf. Corollary 3.6 and Example 3.7. Theorem 4.2. Let X be a finite-dimensional real smooth normed space. Then there exists an ε < 1 such that the Birkhoff orthogonality is ε-symmetric. Proof. Assume the converse. Then there exist an increasing sequence εn such that εn 1 and sequences of unit vectors xn , yn such that xn ⊥B yn
and yn ⊥ εBn xn .
The latter condition can be equivalently written as: εn < ϕn (xn ) ,
∀ ϕn ∈ J (yn ) .
We can choose convergent subsequences xnk → x, ynk → y, ϕnk → ϕ (closed balls in X, X ∗ are compact) and get x⊥B y,
ϕ ∈ J(y),
and 1 = ϕ(x).
Since x⊥B y, there exists x∗ ∈ J(x) such that x∗ (y) = 0. Observe that also ϕ(x) = 1, whence ϕ ∈ J(x). Since X is smooth, ϕ = x∗ . Thus 0 = x∗ (y) = ϕ(y) = 1, a contradiction. 2
634
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
In the sequel we consider an approximate orthogonality in various spaces (X, X ∗ and X ∗∗ ). However, we use one common symbol ⊥εB . For a real normed space X we denote by π : X → X ∗∗ the canonical isometric embedding of X into X ∗∗ : π(x) : X ∗ x∗ → x∗ (x) ∈ R. Since π is a linear isometry from X onto π(X), for arbitrary x, y ∈ X and ε ∈ [0, 1) we have x⊥εB y ⇐⇒ π(x)⊥εB π(y).
(4.1)
Theorem 4.3. Let X be a real uniformly convex Banach space. Then, there exists ε ∈ [0, 1) (the same as in Theorem 4.1) such that the Birkhoff orthogonality in X ∗∗ is ε-symmetric. Proof. Uniform convexity in a Banach space implies reflexivity, thus for arbitrary x∗∗ , y ∗∗ ∈ X ∗∗ we have x∗∗ = π(x) and y ∗∗ = π(y) for some x, y ∈ X. Applying (4.1) and Theorem 4.1 we get x∗∗ ⊥B y ∗∗ ⇔ π(x)⊥B π(y) ⇔ x⊥B y ⇒ y⊥εB x ⇔ π(y)⊥εB π(x) ⇔ y ∗∗ ⊥εB x∗∗ , which completes the proof. 2 Suppose now that X is reflexive, smooth and strictly convex. In particular, smoothness guarantees that for each x ∈ X there is a unique supporting functional, denoted by ϕx (that is J(x) = {ϕx }). Moreover, the mapping SX x → ϕx ∈ SX ∗ is injective. Indeed, if ϕx = ϕy , then ϕx (x + y) = ϕx (x) + ϕx (y) = ϕx (x) + ϕy (y) = 1 + 1 = 2. Thus 2 = ϕx (x + y) ≤ x + y ≤ 2, whence x + y = 2 and strict convexity yields x = y. Lemma 4.4. Let X be a real reflexive, strictly convex and smooth Banach space. Then, for any ε ∈ [0, 1) x⊥εB y ⇐⇒ ϕy ⊥εB ϕx ,
∀ x, y ∈ SX .
(4.2)
Proof. Let x, y ∈ SX . It follows then from (1.3) x⊥εB y ⇐⇒ |ϕx (y)| ≤ ε.
(4.3)
On the other hand, we have from (1.3) applied to X ∗ : ϕy ⊥εB ϕx ⇐⇒ ∃ Φ ∈ X ∗∗ s.t. Φ = 1, Φ(ϕy ) = 1, and |Φ(ϕx )| ≤ ε. Reflexivity of X yields that for some a ∈ X, Φ = π(a). Moreover, a = π(a) = Φ = 1 and ϕy (a) = π(a)(ϕy ) = Φ(ϕy ) = 1, which means that ϕy ∈ J(a) = {ϕa }. It follows that ϕy = ϕa , hence a = y. Moreover, |ϕx (y)| = |ϕx (a)| = |π(a)(ϕx )| = |Φ(ϕx )| ≤ ε. Thus we have ϕy ⊥εB ϕx ⇐⇒ |ϕx (y)| ≤ ε. Now (4.2) follows from (4.3) and (4.4).
(4.4)
2
Theorem 4.5. Let X be a real, reflexive, strictly convex and smooth Banach space. If the orthogonality relation ⊥B in X is ε-symmetric (with some ε ∈ [0, 1)), then the orthogonality relation in X ∗ is also ε-symmetric (with the same ε).
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
635
Proof. Assume ε-symmetry of ⊥B in X. Let x∗ , y ∗ ∈ X ∗ and x∗ ⊥B y ∗ . Since the relations ⊥B and ⊥εB are homogeneous, without loss of generality we assume that x∗ , y ∗ ∈ SX ∗ . Reflexivity of X yields that x∗ , y ∗ attain their norms, i.e., there exist x, y ∈ SX such that x∗ ∈ J(x), y ∗ ∈ J(y). Since X is smooth, x∗ = ϕx and y ∗ = ϕy and applying Lemma 4.4 twice and ε-symmetry of ⊥B in X, we get x∗ ⊥B y ∗ ⇔ ϕx ⊥B ϕy ⇔ y⊥B x ⇒ x⊥εB y ⇔ ϕy ⊥εB ϕx ⇔ y ∗ ⊥εB x∗ , which completes the proof. 2 Finally, applying Theorems 4.1, 4.5 and 4.3, we conclude. Theorem 4.6. Let X be a real uniformly convex and smooth Banach space. Then, there exists ε ∈ [0, 1) such that the Birkhoff orthogonality relations in X, X ∗ and X ∗∗ are ε-symmetric (with the same ε). 5. Birkhoff orthogonality symmetry constant Let X be a normed space with dim X ≥ 2. For X we define a constant connected with the symmetry of the Birkhoff orthogonality: S(X) := inf{ε ∈ [0, 1] : ∀ x, y ∈ X x⊥B y ⇒ y⊥εB x}. Remark 2.2 yields that, in fact, the infimum in the above definition can be replaced by minimum. Obviously, S(X) ∈ [0, 1] and S(X) = 0 means that ⊥B is symmetric, i.e., X is a Radon plane or an inner product space. S(X) = 1 means that ⊥B is not ε-symmetric with any ε ∈ [0, 1), i.e., ⊥B is not approximately symmetric. Bearing in mind the results obtained in previous sections of the paper, we can immediately formulate some further properties of S(X). • From Theorem 3.8 we get for an arbitrary real normed space X: S(X) ≤ 1 − 2δX (1) ≤ 1.
(5.1)
It follows that if δX (1) > 0 (in particular if X is uniformly convex), then S(X) < 1. The reverse is not true. For, let consider a finite-dimensional smooth space X with R(X) ≥ 1 (cf. Example 3.7 and the space Xa with a > 1). Then δX (1) = 0 whereas it follows from Theorem 4.2 that S(X) < 1. • If X satisfies (3.4), Theorem 3.4 yields 1 R(X) ≤ S(X). 2
(5.2)
The assumption (3.4) is essential, cf. Example 3.5. • If X is a real uniformly convex and smooth Banach space, then applying Theorem 4.6 we get S(X) = S(X ∗ ) = S(X ∗∗ ) < 1.
(5.3)
Proposition 5.1. Let X be a normed space. Then S(X) = sup{S(X0 ) : X0 is a two-dimensional subspace of X}.
(5.4)
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
636
Proof. Denote S2 (X) := sup{S(X0 ) : X0 is a two-dimensional subspace of X}. We will show that S2 (X) = S(X). Since X0 ⊂ X, S(X0 ) ≤ S(X), whence S2 (X) ≤ S(X). If S2 (X) = 1, then S(X) = 1. Assume S2 (X) < 1. For n ∈ N such that S2 (X) + n1 ≤ 1 let ε ∈ S2 (X), S2 (X) + n1 . Let x, y ∈ X, x⊥B y, X0 := Lin{x, y}. There is S(X0 ) ≤ S2 (X) < ε < S2 (X) + n1 . Since ε > S(X0 ) and x, y ∈ X0 , we have y⊥εB x. Thus we have proved x⊥B y ⇒ y⊥εB x whence S(X) ≤ ε < S2 (X) + n1 . Letting n → ∞, S(X) ≤ S2 (X) and finally S(X) = S2 (X). 2 Example 5.2. Let X = R3 with the norm (x, y, z) = max{ x2 + y 2 , |z|},
∀ (x, y, z) ∈ R3 .
Let X1 = {(x, y, 0) : x, y ∈ R}
and X2 = {(0, y, z) : y, z ∈ R}.
Then S(X1 ) = 0, S(X2 ) = 1 and S(X) = 1. Under some additional assumptions we will improve the estimation (5.1). Theorem 5.3. Let (X, · ) be a real, uniformly smooth Banach space. Suppose that · 2 is an equivalent norm in X: ax ≤ x2 ≤ bx,
∀x ∈ X
(5.5)
2 (with some 0 < a ≤ b) such that H := (X, · 2 ) is a Hilbert space. Finally, assume that 1 − 2δX ∗ (1) < ab . Then
2 1 − ab −1 . (5.6) S(X) ≤ δX ∗ 2 Proof. Fix x, y ∈ X and assume x⊥B y. Then for any λ ∈ R we have a x2 ≤ ax ≤ ax + λy ≤ x + λy2 b and it follows that x⊥ B y in H (cf. (1.5)) with δ := 1 − ab . Since H is a Hilbert space, the approximate δ orthogonality is symmetric whence we have also y⊥ B x. It gives δ
a y2 ≤ y + λx2 , b and together with (5.5) a 1 a2 y ≤ 2 y2 ≤ y + λx2 ≤ y + λx. 2 b b b Defining γ := 1 −
a 2
we have (1 − γ)y ≤ y + λx, i.e., y⊥ B x in X. Applying Theorem 1.4 (notice that γ −1 γ γ < 2δX ∗ (1)) we obtain y⊥ηB x with η = δX . ∗ 2 Since x, y were chosen arbitrarily, we have proved that the orthogonality relation ⊥B in X is η-symmetric. Thus, in particular, S(X) ≤ η. 2 b
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
637
Let d(X, Y ) denote the Banach–Mazur distance of isomorphic spaces X and Y , i.e., d(X, Y ) := inf{T T −1 : T : X → Y is a linear isomorphism onto Y } ≥ 1. If the estimation (5.5) is sharp, then d(H, X) =
b a
and Theorem 5.3 can be reformulated as follows.
Theorem 5.4. Let H be a Hilbert space and let X be a uniformly smooth Banach space. Assume d(H, X) ≤
1 1 − 2δX ∗ (1)
.
Then
S(X) ≤
−1 δX ∗
1−
1 d(H,X)2
2
.
If a Banach space X is uniformly smooth, then it is reflexive and X ∗ is uniformly convex. Thus −1 δX ∗ : [0, 2] → [0, 1] is a continuous increasing bijection and hence δX ∗ : [0, 1] → [0, 2] is also continuous, increasing and bijective. Our previous theorem yields that if d(H, X) → 1+ (for some Hilbert space H), then S(X) → 0 = S(H). It means some kind of continuity of X → S(X). However, the following example shows that in general this correspondence is not continuous. Example 5.5. Let X = R2 with an l1 -l∞ norm, i.e., with the unit sphere being a hexagon with vertices at x1 = (1, 0), x2 = (0, 1), x3 = (−1, 1), x4 = (−1, 0), x5 = (0, −1), x6 = (1, −1). For an arbitrary positive constant δ let Y = Yδ = R2 with the norm such that the unit sphere is a hexagon with vertices at y1 = (1, 0), y2 = (0, 1), y3 = (−1 − δ, 1 − δ), y4 = (−1, 0), y5 = (0, −1), y6 = (1 + δ, −1 + δ). See Fig. 4.
Fig. 4. Example 5.5, unit spheres in X and Y .
Let T : X → Y be a linear mapping defined by T x2 = y2 and T x3 = y3 , whence T (x, y) = ((1 + δ)x, δx + y),
∀ (x, y) ∈ R2
and T
−1
(x, y) =
δ 1 x, y − x , 1+δ 1+δ
∀ (x, y) ∈ R2 .
638
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
1+δ 1+δ One can notice that T (BX ) ⊂ 1−δ BY and T −1 (BY ) ⊂ BX . Therefore T ≤ 1−δ and T −1 ≤ 1 and −1 hence T T → 1 as δ → 0. It means that, for an arbitrary η > 0 and with sufficiently small δ, we have d(X, Y ) < 1 + η. The space X is a Radon plane, therefore S(X) = 0. For the space Y we can see that R(Y ) > 1. Moreover, the condition (3.4) is satisfied. Thus it follows from (5.2) that
1 1 < R(Y ) ≤ S(Y ). 2 2 To conclude, for the space X we have S(X) = 0 and for an arbitrary η > 0 there exists a normed space Y such that d(X, Y ) < 1 + η, and S(Y ) > 12 . In the following example we give some estimation for S(lnp ). Example 5.6. Suppose that p, q > 1 are conjugated (i.e., p1 + 1q = 1). Consider the space lnp and its dual lnq . It is known (cf. [11, Lemma 2]) that the Banach–Mazur distance between lnp and ln2 is equal to: d(lnp , ln2 )
d :=
1 1
p−2
=n
(consequently d(lnq , ln2 ) = d(lnp , ln2 ) = d). Suppose that p and q are sufficiently close to 2, namely such that
1 1
p−2
n
1 1
q−2
=n
≤ min
1 1 − 2δlp (1)
,
1 1 − 2δlq (1)
Then it follows from Theorem 5.4 S(lnp )
≤
δl−1 q n
1 − d12 2
,
and by monotonicity of δ we have δlnq (S(lnp )) ≤
1 − d12 . 2
Assume now 1 < p ≤ 2. Then q ≥ 2 and we have (cf. [12, Theorem 2]) ε q q1 δlnq (ε) = 1 − 1 − 2 whence we get 1 1 − d12 S(lnp )q q p q (S(l )) ≤ . 1− 1− = δ l n n 2q 2 Thus, for p ≤ 2 and such that (5.7) holds, we have S(lnp )
≤
1 2 − 1+ 2 d q
q q1 .
.
(5.7)
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
639
If p ≥ 2, then q ≤ 2 and interchanging the roles of p and q we get S(lnq )
p p1 1 p ≤ 2 − 1+ 2 . d
But S(lnq ) = S(lnp ) (see (5.3)), so we get S(lnp )
2q − 1 + ≤ 2p − 1 +
1 1 q q d2 1 1 p p d2
,
1 < p ≤ 2,
, p ≥ 2.
It follows that for any p > 1 satisfying (5.7) we have S(lnp )
≤ max
1 2 − 1+ 2 d q
q q1 p p1 1 p , 2 − 1+ 2 . d
Notice that if p → 2, then q → 2 and d → 1. Therefore the right hand side of the above estimation tends to zero, which means limp→2 S(lnp ) = S(ln2 ). Let X be a real Banach space and let Y be a closed subspace of X. A continuous linear operator P : X → Y is called a projection onto Y if P |Y = Id|Y . By P(X, Y ) we denote the family of all projections from X to Y . If P(X, Y ) = ∅, we define a relative projection constant of Y with respect to X as λ(Y, X) := inf{P : P ∈ P(X, Y )}. If there exists a projection P : X → Y satisfying P = λ(Y, X), we call P a minimal projection. In particular, if dim X < ∞, then any subspace Y admits a (not necessarily unique) minimal projection. Cf. [20] for basic information on this topic. The case λ(Y, X) = 1 attracts much attention (cf. a survey [21]); Y is called then a one-complemented subspace of X. Another problem is to give a quantitative estimation of the relative projection constant. The classical result of Bohnenblust is quoted below. Recent results, as well as more information on the problem, can be found, e.g., in [17,18]. Theorem 5.7 ([4]). Let X be an n-dimensional normed space and let Y be a subspace of X with dim Y = n−1. Then, λ(Y, X) ≤ 2 − n2 . If the Birkhoff orthogonality is ε-symmetric, the above estimation can be improved. Theorem 5.8. Let X be a real reflexive space for which ⊥B is ε-symmetric with some ε ∈ 0, 12 . Let Y be a 1 closed subspace with codim Y = 1. Then λ(Y, X) ≤ 1−2ε . Proof. Assuming ε-symmetry of ⊥B and applying (1.6), we get x⊥B y
⇒
y⊥ B x. 2ε
(5.8)
Let Y be a closed subspace of X with codim Y = 1. Since X is reflexive, there exists x ∈ SX such that x⊥B Y (see [16] or [1, Theorem 4.13]). Applying (5.8), we get Y ⊥ B x. 2ε
(5.9)
640
J. Chmieliński, P. Wójcik / J. Math. Anal. Appl. 461 (2018) 625–640
For each vector y + αx ∈ X = Y + Lin{x} (unique decomposition) we define P (y + αx) := y. Then P ∈ P(X, Y ) and, by (5.9), (1 − 2ε)P (y + αx) = (1 − 2ε)y ≤ y + αx. Thus for each y + αx ∈ X we have P (y + αx) ≤ 1 we get λ(Y, X) ≤ 1−2ε . 2
1 1−2ε y + αx,
whence P ≤
1 1−2ε .
Since λ(Y, X) ≤ P ,
If X is an n-dimensional normed space and the orthogonality relation ⊥B is ε-symmetric with ε < 12 − 4−1 4 , n then the estimation of λ(Y, X) obtained in Theorem 5.8 is better than in Theorem 5.7. Moreover, the estimation in Theorem 5.8 does not depend on n. Finally, as opposed to Theorem 5.7, Theorem 5.8 is valid without any restriction on the dimension of the normed space X. As a corollary we get another estimation of the symmetry constant S(X). Theorem 5.9. Let X be a real reflexive normed space and let Y be a closed subspace of X with codim Y = 1. Then 1 1 1− . S(X) ≥ 2 λ(Y, X) References [1] J. Alonso, H. Martini, S. Wu, On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces, Aequationes Math. 83 (2012) 153–189. [2] C. Alsina, J. Sikorska, M.S. Tomás, Norm Derivatives and Characterizations of Inner Product Spaces, World Scientific, Hackensack, NJ, 2010. [3] G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J. 1 (1935) 169–172. [4] F. Bohnenblust, Convex regions and projections in Minkowski spaces, Ann. of Math. 39 (1938) 301–308. [5] J. Chmieliński, On an ε-Birkhoff orthogonality, J. Inequal. Pure Appl. Math. 6 (3) (2005) 79. [6] J. Chmieliński, Operators reversing orthogonality in normed spaces, Adv. Oper. Theory 1 (1) (2016) 8–14. [7] J. Chmieliński, P. Wójcik, ρ-orthogonality and its preservation-revisited, in: Recent Developments in Functional Equations and Inequalities, in: Banach Center Publications, vol. 99, 2013, pp. 17–30. [8] J. Chmieliński, T. Stypuła, P. Wójcik, Approximate orthogonality in normed spaces and its applications, Linear Algebra Appl. 531 (2017) 305–317. [9] S.S. Dragomir, On approximation of continuous linear functionals in normed linear spaces, An. Univ. Timiş. Ser. Ştiinţ. Mat. 29 (1991) 51–58. [10] P.M. Gruber, Stability of Blaschke’s characterization of ellipsoids and Radon norms, Discrete Comput. Geom. 17 (1997) 411–427. [11] V.E. Gurarii, M.I. Kadec, V.E. Macaev, On the distance between isomorphic spaces of finite dimension, Math. Sb. 70 (1966) 481–489 (in Russian). [12] O. Hanner, On the uniform convexity of Lp and lp , Ark. Mat. 3 (1956) 239–244. [13] S. Heinrich, The differentiability of the norm in spaces of operators, Funct. Anal. Appl. 9 (1976) 360–362; transl. from Funktsional. Anal. i Prilozhen. 9 (4) (1975) 93–94. [14] R.C. James, Orthogonality in normed linear spaces, Duke Math. J. 12 (1945) 291–301. [15] R.C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947) 265–292. [16] R.C. James, Reflexivity and the sup of linear functionals, Israel J. Math. 13 (1972) 289–300. [17] T. Kobos, Hyperplanes of finite-dimensional normed spaces with the maximal relative projection constant, Bull. Aust. Math. Soc. 91 (2015) 447–463. [18] T. Kobos, A uniform estimate of the relative projection constant, J. Approx. Theory 225 (2018) 58–75. [19] B. Mojškerc, A. Turnšek, Mappings approximately preserving orthogonality in normed spaces, Nonlinear Anal. 73 (2010) 3821–3831. [20] W. Odyniec, G. Lewicki, Minimal Projections in Banach Spaces, Lecture Notes in Mathematics, vol. 1449, Springer, Berlin, Heidelberg, New York, 1990. [21] B. Randrianantoanina, Norm-one projections in Banach spaces, Taiwanese J. Math. 5 (2001) 35–95. [22] T. Stypuła, P. Wójcik, Characterizations of rotundity and smoothness by approximate orthogonalities, Ann. Math. Sil. 30 (2016) 183–201. [23] P. Wójcik, Norm-parallelism in classical M-ideals, Indag. Math. 28 (2017) 287–293.