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Approximate orthogonality in normed spaces and its applications
Accepted Manuscript Approximate orthogonality in normed spaces and its applications
Jacek Chmieli´nski, Tomasz Stypuła, Paweł Wójcik
PII: DOI: Reference:
S0024-3795(17)30363-4 http://dx.doi.org/10.1016/j.laa.2017.06.001 LAA 14199
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Linear Algebra and its Applications
Received date: Accepted date:
1 March 2017 1 June 2017
Please cite this article in press as: J. Chmieli´nski et al., Approximate orthogonality in normed spaces and its applications, Linear Algebra Appl. (2017), http://dx.doi.org/10.1016/j.laa.2017.06.001
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Approximate orthogonality in normed spaces and its applications Jacek Chmieli´ nski, Tomasz Stypula, Pawel W´ojcik Department of Mathematics, Pedagogical University of Cracow Podchor¸az˙ ych 2, 30-084 Krak´ ow, Poland
Abstract In a normed space we consider an approximate orthogonality relation related to the Birkhoff orthogonality. We give some properties of this relation as well as applications. In particular, we characterize the approximate orthogonality in the class of linear bounded operators on a Hilbert space. Keywords: Birkhoff orthogonality; approximate Birkhoff orthogonality; approximate orthogonality of operators 2010 MSC: 46B20, 46B28, 46C05, 46C50, 47L25, 47L30 1. Introduction In an inner product space (X, ·|·), with the standard orthogonality relation x⊥y ⇔ x|y = 0, a natural way to define an approximate orthogonality (or more precisely ε-orthogonality with ε ∈ [0, 1)) is by: x⊥ε y
⇐⇒
| x|y | ≤ ε x y ,
x, y ∈ X.
It is easy to show, in this setting, the following characterization: x⊥ε y
⇐⇒
∃ z ∈ X : x⊥z, z − y ≤ ε y .
(1.1)
x + y for x = 0 and Indeed, if x⊥ε y, then it is enough to take z = − x|y x2 z = y for x = 0. Conversely, assuming x⊥z and z − y ≤ ε y , we get | x|y | = | x|y − z | ≤ x y − z ≤ ε x y , that is x⊥ε y. Notice that z Email addresses: [email protected] (Jacek Chmieli´ nski), [email protected] (Tomasz Stypula), [email protected] (Pawel W´ ojcik) Preprint submitted to Linear Algebra and Applications
June 2, 2017
was actually chosen from Lin {x, y}. In the subsequent section we will extend the above characterization to the case of Birkhoff orthogonality in a normed space. Then, we will show its usefulness in further considerations connected with the approximate Birkhoff orthogonality, as well as in some applications, in particular, in the operator theory. Throughout the paper we will consider a normed space (X, · ), with dim X ≥ 2, over the scalar field K ∈ {R, C} (which sometimes will be restricted to R). X ∗ denotes the dual space for X; BX and SX will stand for the closed unit ball and the unit sphere, respectively, in X. By ExtA we will mean the set of extremal points of A. 2. Approximate Birkhoff orthogonality 2.1. Birkhoff orthogonality In a normed space X we consider the Birkhoff orthogonality: x⊥B y
⇐⇒
∀λ ∈ K :
x + λy ≥ x
(cf. [6, 13, 14] or a survey [2]). There are numerous characterizations of this relation which refer to various geometrical properties connected with the concept of orthogonality. For x ∈ X let J(x) denote the (nonempty) set of its supporting functionals: J(x) = {ϕ ∈ X ∗ :
ϕ = 1, ϕ(x) = x }.
A well known characterization of the Birkhoff orthogonality is due to James (cf. [14, Corollary 2.2]): x⊥B y
⇐⇒
∃ϕ ∈ J(x) : ϕ(y) = 0.
(2.1)
Another characterization is due to Singer [21]. For a real normed space X and x, y ∈ X we have: x⊥B y
⇐⇒
∃ϕ, ψ ∈ Ext(BX ∗ ) ∩ J(x) ∃α ∈ [0, 1] : αϕ(y) + (1 − α)ψ(y) = 0.
2
(2.2)
2.2. Approximate Birkhoff orthogonality The approximate Birkhoff orthogonality (ε-Birkhoff orthogonality with ε ∈ [0, 1)) can be defined by x⊥εB y
⇐⇒
∀λ ∈ K :
x + λy 2 ≥ x 2 − 2ε x λy
as introduced in [7]. It is easy to see that for ε = 0, ⊥εB = ⊥B and for inner product spaces we have ⊥εB = ⊥ε . Later on, we will give some characterizations of ⊥εB related to (2.1) and (2.2) (Theorems 2.3 and 2.4). We need to recall some related concepts. First are the norm derivatives (cf. [3] for the background and properties of these functionals): ρ± (x, y) = lim± t→0
x + ty 2 − x 2 , 2t
x, y ∈ X, t ∈ R
and related to them orthogonality relations: x⊥ρ± y
ρ± (x, y) = 0.
⇐⇒
The corresponding approximate orthogonalities (defined and investigated in [8, 9]) are: x⊥ερ± y ⇐⇒ |ρ± (x, y)| ≤ ε x y . We recall (cf. [16, 11] or the monograph [10]) that each norm in X admits a semi–inner product (generating this norm), that is a functional [·|·] : X × X → K satisfying the conditions: • [λx + μy|z] = λ [x|z] + μ [y|z] , x, y, z ∈ X, λ, μ ∈ K; • [x|λy] = λ [x|y] , x, y ∈ X, λ ∈ K; • | [x|y] | ≤ x y , x, y ∈ X; • [x|x] = x 2 , x ∈ X. Generally, unless the norm is smooth, there could be many different semi– inner products related to a given norm in X. If the norm in X comes from an inner product, then the inner product itself is the unique semi-inner product. For a fixed semi-inner product [·|·] and x, y ∈ X, we define the semi– orthogonality x⊥s y ⇐⇒ [y|x] = 0 3
and the ε-semi–orthogonality (approximate semi–orthogonality) x⊥εs y
⇐⇒
| [y|x] | ≤ ε x y .
For any x, y ∈ X with x⊥B y there exists a semi-inner product [·|·] such that [y|x] = 0 (cf. [10]). It is also known [9, Theorem 3.2] that if X is a real normed space and ε ∈ [0, 1), then ⊥εs ⊂ ⊥εB , ⊥ερ+ ⊂ ⊥εB and ⊥ερ− ⊂ ⊥εB . Moreover, in a smooth space all the relations coincide, that is ⊥εs = ⊥ερ+ = ⊥ερ− = ⊥εB (cf. also [22]). Let SIP X denote the set of all semi–inner products in X. It is known (cf. [10]) that for a real normed space X
and
ρ+ (x, y) = sup{[y|x] : [·|·] ∈ SIP X }
(2.3)
ρ− (x, y) = inf{[y|x] : [·|·] ∈ SIP X }.
(2.4)
We will also use the following characterization of the ε-Birkhoff orthogonality (cf. [9, Theorem 3.1]): x⊥εB y
⇐⇒
ρ− (x, y) − ε x y ≤ 0 ≤ ρ+ (x, y) + ε x y .
(2.5)
2.3. New characterizations of the approximate Birkhoff orthogonality We start with an auxiliary result. Proposition 2.1. Let X be a real normed space and let x, y ∈ X be fixed, such that x⊥εB y (with some ε ∈ [0, 1)). Then, for each n ∈ N there exists a semi-inner product [·|·]n in X such that 1
x y . (2.6) | [y|x]n | ≤ ε + n Proof. Applying (2.5) and (2.3)-(2.4), for each integer n such that n1 ∈ (0, 1− ε) we may choose semi-inner products [·|·]n and [·|·]n such that 1 1 [y|x]n < ε + (2.7)
x y and − ε +
x y < [y|x]n . n n It follows that for some λn ∈ [0, 1] we have 1 1
x y ≤ λn [y|x]n + (1 − λn ) [y|x]n ≤ ε +
x y . − ε+ n n 4
(We use a simple fact that if a < δ and −δ < b for some a, b ∈ R and δ > 0, then −δ ≤ λa + (1 − λ)b ≤ δ for some λ ∈ [0, 1].) Now, we consider a semi-inner product [·|·]n := λn [·|·]n + (1 − λn ) [·|·]n and from the above inequalities it follows 1 1 − ε+
x y ≤ [y|x]n ≤ ε +
x y . n n
We are in position to prove the main result of this section. Theorem 2.2. Let X be a real normed space. For x, y ∈ X and ε ∈ [0, 1): x⊥εB y
⇐⇒
∃z ∈ Lin {x, y} : x⊥B z, z − y ≤ ε y .
(2.8)
Moreover, the implication “⇐” is true also for complex spaces. Proof. First, suppose that x⊥εB y. We assume that x = 0 (otherwise the result is trivial). It follows from Proposition 2.1 that for an arbitrary n ∈ N there exists a semi–inner product [·|·]n in X such that (2.6) holds. Let ⊥s,n denote the orthogonality relation related to this semi–inner product. We [y|x] define zn := − x2n x + y ∈ Lin {x, y} and it is easy to see that x⊥s,n zn . Since ⊥s,n ⊂ ⊥B , it follows x⊥B zn . Applying (2.6) we estimate zn − y and we get 1
y . (2.9) and
zn − y ≤ ε + x⊥B zn n Notice that zn ≤ 2 y which means that the elements of the sequence (zn )n=1,2,... belong to a closed ball in a two-dimensional space Lin {x, y}. Thus there exists z ∈ Lin {x, y} and a subsequence (znk ) convergent to z. Finally, (2.9) and continuity of the norm yield x⊥B z and z − y ≤ ε y . Now, let X be a real or complex normed space and assume that there exists z ∈ X such that x⊥B z and z − y ≤ ε y . Since x⊥B z, there exists a semi-inner product [·|·] such that [z|x] = 0 and thus | [y|x] | = | [y − z|x] | ≤ y − z x ≤ ε y x , i.e., x⊥εs y. Since ⊥εs ⊂ ⊥εB , we get x⊥εB y.
5
Validity of the implication “⇒” in a complex space remains an open problem. We apply the above result to give another characterization of ⊥εB . Such a characterization was already noticed in [17] for smooth spaces; we will prove it now without this restriction. Theorem 2.3. Let X be a real normed space. For x, y ∈ X and ε ∈ [0, 1): x⊥εB y
⇐⇒
∃ ϕ ∈ J(x) : |ϕ(y)| ≤ ε y .
(2.10)
Moreover, the implication “⇐” is true also for complex spaces. Proof. Let x⊥εB y. Applying (2.8), there exists z ∈ X such that x⊥B z and
z − y ≤ ε y . Due to (2.1) there exists ϕ ∈ J(x) such that ϕ(z) = 0. Therefore, |ϕ(y)| = |ϕ(z) − ϕ(y)| ≤ z − y ≤ ε y , as claimed. For the reverse assume that ϕ ∈ J(x) and |ϕ(y)| ≤ ε y . Fix arbitrarily λ ∈ K. We have then |ϕ(λy)| ≤ ε λy and it follows
x + λy 2 ≥ |ϕ(x + λy)|2 ≥ | |ϕ(x)| − |ϕ(λy)| |2 = |ϕ(x)|2 − 2|ϕ(x)| |ϕ(λy)| + |ϕ(λy)|2 ≥ x 2 − 2ε x λy , which means that x⊥εB y. Proceeding similarly, we generalize Singer’s result (2.2). The following theorem will be useful later (cf. the proof of Theorem 3.6). Theorem 2.4. Let X be a real normed space, let x, y ∈ X, and let ε ∈ [0, 1). Then x⊥εB y
⇐⇒
∃ϕ, ψ ∈ Ext(BX ∗ ) ∩ J(x) ∃α ∈ [0, 1] : |αϕ(y) + (1 − α)ψ(y)| ≤ ε y .
(2.11)
Moreover, the implication “⇐” is true also for complex spaces. Proof. Let x⊥εB y. Applying (2.8), there exists z ∈ X such that x⊥B z and
z − y ≤ ε y and by (2.2) there exist ϕ, ψ ∈ Ext(BX ∗ ) ∩ J(x) and α ∈ [0, 1] such that αϕ(z) + (1 − α)ψ(z) = 0. Therefore, |αϕ(y) + (1 − α)ψ(y)| = |αϕ(y − z) + (1 − α)ψ(y − z)| ≤ α z − y + (1 − α) z − y
= z − y ≤ ε y . 6
For the reverse implication fix arbitrarily λ ∈ K. The functional αϕ + (1 − α)ψ is also in J(x). We have then |(αϕ + (1 − α)ψ)(λy)| ≤ ε λy and similarly as before
x + λy 2 ≥ |(αϕ + (1 − α)ψ)(x + λy)|2 ≥ | |(αϕ + (1 − α)ψ)(x)| − |(αϕ + (1 − α)ψ)(λy)| |2 ≥ x 2 − 2ε x λy , i.e., x⊥εB y. Remark 2.5. Since the (approximate) orthogonality relation in an inner product space is symmetric, we can change the roles of x and y and rewrite (1.1) in an equivalent form: x⊥ε y
⇐⇒
∃ z ∈ X : z⊥y, z − x ≤ ε x .
(2.12)
One may ask whether this characterization remains true in normed spaces with the Birkhoff orthogonality. We show that the answer is negative and actually none of the implications in (2.12) is valid in normed spaces, generally. Example 2.6. Consider the space R2 with the maximum norm
(x1 , x2 ) ∞ := max{|x1 |, |x2 |}. Let x = (1, 1), y = (ε, 1) with ε ∈ (0, 1). Since ρ+ (x, y) = 1 and ρ− (x, y) = ε (cf. [3, Example 2.1.2]) the condition (2.5) yields x⊥εB y follows. Let z be such that z⊥B y. If z ∞ = 1, then z = (1, −1) or z = (−1, 1). Thus it follows that z⊥B y ⇔ z = α(−1, 1). Therefore for each z such that z⊥B y the following inequality holds:
z − x ∞ = max{|1 + α|, |1 − α|} ≥ 1 > ε x ∞ . Now, let x = (1, 1 − ε), y = (1, 0), z = (1 − ε, 1) with ε ∈ (0, 1). Then we have z⊥B y and z − x ∞ ≤ ε x ∞ . On the other hand, ρ− (x, y) = 1, thus ρ− (x, y) − ε x ∞ y ∞ > 0, whence x⊥εB y cannot be true. 3. Applications In this part of the paper we present some applications of the main result from Section 2 in particular normed spaces. 7
3.1. Characterization of approximate orthogonality in L(H) Let L(H) be the space of all linear and bounded operators on a Hilbert space H and let K(H) stand for a subspace of L(H) consisting of all compact operators. For a given T ∈ L(H), we denote MT := {x ∈ SH : T x = T } — the set of points from the unit sphere at which T attains its norm (we follow the notation used in [20, Definition 1.1]). Generally, MT may be empty; however, compactness of T implies that MT = ∅ and this is the case, in particular, if dim H < ∞. ˇ Bhatia and Semrl [5] gave the following characterization of the Birkhoff orthogonality in L(H). Theorem 3.1 ([5], Theorem 1.1, Remark 3.1). Let H be a Hilbert space and let T, S ∈ L(H). Then, the following conditions are equivalent: (i) T ⊥B S; (ii) ∃ (xn )∞ n=1 ⊂ SH : lim T xn = T ,
n→∞
lim T xn |Sxn = 0.
n→∞
Moreover, if dim H < ∞, then each of the above conditions is equivalent to: (iii) ∃ x0 ∈ MT :
T x0 ⊥Sx0 .
Independently, the characterization (i)⇔(ii) was given also by Paul [18, Lemma 2]. Then, the above result was developed by various authors. It is known that the equivalence (i) ⇔ (iii) cannot be extended to L(X) for a finite-dimensional normed space X — a counterexample was given by Li and Schneider [15]. Then Ben´ıtez, Fern´andez and Soriano [4] showed that this equivalence is valid if and only if X is a Hilbert space. Other generalizations or applications of Theorem 3.1 have been obtained, e.g., by Arambaˇsi´c and Raji´c [1], Sain, Paul and Hait [19, 20], Grover [12] and recently by W´ojcik [23]. All these papers, however, dealt with the exact orthogonality of operators. ˇ We will provide some extensions of the result of Bhatia and Semrl by giving a characterization of the approximate orthogonality in L(H). Moreover, in the sequel, we will replace the assumption of finite-dimensionality of H by the compactness of the first operator. Theorem 3.2. Let H be a real Hilbert space, let T, S ∈ L(H) and let ε ∈ [0, 1). Then, the following conditions are equivalent: 8
(1) T ⊥εB S; (2) ∃ (xn )∞ n=1 ⊂ SH : lim T xn = T ,
n→∞
lim | T xn |Sxn | ≤ ε T S .
n→∞
Moreover, if dim H < ∞, then each of the above conditions is equivalent to: (3) ∃ x0 ∈ MT :
| T x0 |Sx0 | ≤ ε T S .
If dim H < ∞ and, additionally, MT ⊂ MS , then each of the above three conditions is equivalent also to: (4) ∃ x0 ∈ MT :
T x0 ⊥ε Sx0 .
Proof. (1) ⇒ (2). Assume T ⊥εB S. It follows from (2.8) that for some U ∈ L(H) we have T ⊥B U and U − S ≤ ε S . Then Theorem 3.1 yields that there exists a sequence (xn ) in SH such that T xn → T and T xn |U xn → 0 as n → ∞. Therefore, | T xn |Sxn | ≤ | T xn |U xn | + | T xn |(S − U )xn | ≤ | T xn |U xn | + T S − U
≤ | T xn |U xn | + ε T S . Without loss of generality the sequence (| T xn |Sxn |)∞ n=1 may be assumed convergent — otherwise we choose an appropriate subsequence (xnk ) of (xn ). Letting n → ∞, we get the desired inequality. (2) ⇒ (1). Assume (2). For an arbitrary scalar λ and each n ∈ N we have
T + λS 2 ≥ T xn + λSxn 2 ≥ T xn 2 + 2λ T xn |Sxn ≥ T xn 2 − 2|λ| | T xn |Sxn |. Letting n → ∞ we get T + λS 2 ≥ T 2 − 2ε T λS , whence T ⊥εB S. Assuming (2) and dim H < ∞, the sequence (xn ) must contain a subsequence convergent to some x0 ∈ SH . Continuity of T and S yields (3). The reverse implication is obvious with xn = x0 for n = 1, 2, . . .. Finally if MT ⊂ MS , then for x0 ∈ MT we have T (x0 ) = T and S(x0 ) = S . Thus (3) is equivalent to (4). 9
Remark 3.3. Notice that implication (4)⇒(1) does not require any additional assumptions. Indeed, assuming (4), for an arbitrary λ we have
T + λS 2 ≥ T x0 + λSx0 2 ≥ T x0 2 − 2ε T x0 λSx0
= T 2 − 2ε T λSx0 ≥ T 2 − 2ε T λS , i.e., T ⊥εB S. However, for (3)⇒(4) the assumption MT ⊂ MS is essential. Indeed, let H be a two-dimensional Hilbert space. Fix ε ∈ [0, 1). Let x, y, z ∈ SH be such that x|y = 2ε and x⊥z. Define T, S ∈ L(H) by: T x := x, T z := 0;
Sx := y, Sz := 2z.
Thus T = 1, T x = 1, S ≥ 2, Sx = 1 and x ∈ MT but x ∈ MS . Since | T x|Sx | = | x|y | = 2ε ≤ ε T S , condition (3) is satisfied. On the other hand | T x|Sx | = | x|y | = 2ε > ε T x Sx , which means that T x ⊥ε Sx (actually T x0 ⊥ε Sx0 for all x0 ∈ MT = {x, −x}). Thus (4) is not satisfied. The condition dim H < ∞ in the above theorem can be replaced by the compactness of the operator T . Theorem 3.4. Let H be a real Hilbert space, let T, S ∈ L(H) and let ε ∈ [0, 1). Assume that MT ⊂ MS and T ∈ K(H). Then T ⊥εB S if and only if ∃ x0 ∈ M T :
T x0 ⊥ε Sx0 .
Proof. Let T ⊥εB S. It follows from (2.8) that for an operator U ∈ L(H) we have T ⊥B U and U − S ≤ ε S . Then from Theorem 3.1 we get a sequence (xn ) in SH satisfying T xn → T and T xn |U xn → 0 as n → ∞. Since BH is weakly compact, there exist a subsequence (xnk ) of (xn ) and x0 ∈ BH such that (xnk ) is weakly convergent to x0 . Since T is compact, we have a strong convergence T xnk → T x0 and it follows T x0 = T . Thus x0 ≥ 1 and hence x0 ∈ SH . Notice that | T xnk |U xnk − T x0 |U x0 | ≤ | T xnk |U xnk − T x0 |U xnk | +| T x0 |U xnk − T x0 |U x0 | ≤ T xnk − T x0 U xnk
+| U ∗ T x0 |xnk − U ∗ T x0 |x0 |. 10
The sequence (U xnk ) is bounded, hence T xnk − T x0 U xnk → 0. Since (xnk ) is weakly convergent to x0 , U ∗ T x0 |xnk → U ∗ T x0 |x0 . The right hand side of the above estimation is convergent to zero, thus we have T xnk |U xnk − T x0 |U x0 → 0. Since T xnk |U xnk → 0, we get T x0 |U x0 = 0. We have x0 ∈ MT ⊂ MS , thus S = Sx0 and
U x0 − Sx0 ≤ U − S ≤ ε S = ε Sx0 . Summarizing, we have obtained T x0 ⊥U x0 and U x0 − Sx0 ≤ ε Sx0 , and therefore, due to (2.8), we get T x0 ⊥ε Sx0 . The reverse implication was proved in Remark 3.3. As a corollary we get the following theorem. Theorem 3.5. Let H be a real Hilbert space. Let T ∈ K(H), S ∈ L(H). Assume that MT ⊂ MS . Then T ⊥εB S
=⇒
S⊥εB T.
(3.1)
Proof. For the proof we need to apply Theorem 3.4, the symmetry of the relation ⊥ε in H and Remark 3.3. Of course, if both operators T, S are compact and MT = MS , then T ⊥εB S
⇐⇒
S⊥εB T.
3.2. Characterization of approximate orthogonality Let K be a locally compact topological space. the space of all real continuous mappings defined supremum norm. We consider a subspace C0 (K) of
in C0 (K) As usual, C(K) denotes on K, enowed with the C(K):
C0 (K) := {f ∈ C(K) : ∀ ε > 0, the set {t ∈ K : |f (t)| ≥ ε} is compact}. For f ∈ C0 (K) the set Mf := {t ∈ K : |f (t)| = f } is always nonempty and compact. Theorem 3.6. Let f, g ∈ C0 (K), f = 0 = g. Assume that Mf is connected. Then, the following conditions are equivalent: 11
(a) f ⊥εB g; (b) ∃ t1 ∈ Mf : |g(t1 )| ≤ ε g . Proof. (a)⇒(b). Let f ⊥εB g. It follows from Theorem 2.4 that there exist extremal points a∗ , b∗ ∈ Ext BC0 (K)∗ and α ∈ [0, 1], such that a∗ (f ) = b∗ (f ) =
f and |αa∗ (g) + (1 − α)b∗ (g)| ≤ ε g . (3.2) It is known (cf. [21, Chapter I, §1, 1.10, 1.11]) that extremal functionals a∗ , b∗ in C0 (K) have the form ±ψu , ±ψw for some u, w ∈ Mf , where ψt : C0 (K) → R is defined as ψt (h) := h(t). Since Mf is connected, f takes on Mf values of the same sign. We can assume without loss of generality that f (t) = f for t ∈ Mf . Then a∗ , b∗ take the form ψu , ψw . Thus a∗ (g) = ψu (g) = g(u) and similarly b∗ (g) = ψw (g) = g(w). Hence from the inequality (3.2) we get |αg(u) + (1 − α)g(w)| ≤ ε g .
(3.3)
If |g(u)| ≤ ε g or |g(w)| ≤ ε g , then we take t1 = u or t1 = w, respectively. If g(u) < −ε g and g(w) > ε g (or g(w) < −ε g and g(u) > ε g ) we use the fact that Mf is connected and the Darboux property for g to get for some t1 ∈ Mf that −ε g ≤ g(t1 ) ≤ ε g as required. We eliminate the remaining cases g(u), g(w) > ε g or g(u), g(w) < −ε g as they would contradict (3.3). (b)⇒(a). Since t1 ∈ Mf , for the functional ϕ = ψt1 (or ϕ = −ψt1 ) we have ϕ = 1 and ϕ(f ) = |f (t1 )| = f , hence ϕ ∈ J(f ). Moreover, |ϕ(g)| = |ψt1 (g)| = |g(t1 )| ≤ ε g and (a) follows from (2.10). 3.3. Characterization of approximate orthogonality in quotient spaces Let M be a closed subspace of a normed space X. Let [x] := x + M be the coset of M with respect to x, and let X/M := {[x] : x ∈ X} be the quotient space endowed with the norm
[x] = dist (x, M ) = inf x − m . m∈M
Notice that x⊥B M ⇐⇒ x = [x] , whence it follows that for any m0 ∈ M we have x − m0 ⊥B M ⇐⇒ x − m0 = [x] . (3.4) 12
We say that M is proximinal if and only if ∀ x ∈ X ∃ y ∈ M : x − y = dist (x, M ). A proximinal subspace is closed. Theorem 3.7. Let X be a real smooth normed space and let M ⊂ X be a proximinal subspace of X. For x, y ∈ X the following conditions are equivalent: (1) [x]⊥εB [y]; (2) ∀ m1 , m2 ∈ M such that x − m1 ⊥B M there is: x − m1 ⊥εB y − m2 ; (3) ∃ m1 , m2 ∈ M such that x−m1 ⊥B M , y −m2 ⊥B M and x−m1 ⊥εB y −m2 . Proof. (1)⇒(2). Assume (1) and fix m1 ∈ M such that x − m1 ⊥B M and m2 ∈ M . It follows from (3.4) and from the definition of ε-orthogonality that for any λ we have
x − m1 2 = [x] 2 ≤ [x] + λ[y] 2 + 2ε [x] λ[y]
≤ x − m1 + λ(y − m2 ) 2 + 2ε x − m1 λ(y − m2 ) , which means x − m1 ⊥εB y − m2 . The implication (2)⇒(3) is obvious (here we use proximinality of M ). Now, we prove (3)⇒(1). Let m1 , m2 ∈ M be such that x−m1 ⊥B M , y −m2 ⊥B M and x−m1 ⊥εB y −m2 . It follows from Theorem 2.2 that there exists z ∈ Lin {x − m1 , y − m2 } such that x − m1 ⊥B z and
z − (y − m2 ) ≤ ε y − m2 . It is clear that [z] ∈ Lin {[x], [y]}. Smoothness of X yields that the Birkhoff orthogonality is additive on the right. Thus it follows x − m1 ⊥B z − m for all m ∈ M whence for an arbitrary scalar λ and arbitrary m ∈ M we have
[x] = x − m1 ≤ x − m1 + λ(z − m) . Therefore, for any m ∈ M we have [x] ≤ x + λz − m . Passing to infimum over m ∈ M we obtain [x] ≤ [x + λz] = [x] + λ[z] , which means [x]⊥B [z]. Moreover,
[z] − [y] = [z − y] ≤ z − y + m2 ≤ ε y − m2 = ε [y] , hence applying again Theorem 2.2 we get [x]⊥εB [y]. [1] Lj. Arambaˇsi´c, R. Raji´c, The Birkhoff-James orthogonality in Hilbert C ∗ -modules, Linear Algebra Appl. 437 (2012), 1913–1929. 13
[2] J. Alonso, H. Martini, S. Wu, On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces, Aequationes Math. 83 (2012), 153–189. [3] C. Alsina, J. Sikorska, M.S. Tom´as, Norm Derivatives and Characterizations of Inner Product Spaces, World Scientific, Hackensack, NJ, 2010. [4] C. Ben´ıtez, M. Fern´andez, M.L. Soriano, Orthogonality of matrices, Linear Algebra Appl. 422 (2007), 155–163. ˇ [5] R. Bhatia, P. Semrl, Orthogonality of matrices and some distance problems, Linear Algebra Appl. 287 (1999), 77–85. [6] G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J. 1 (1935), 169–172. [7] J. Chmieli´ nski, On an ε-Birkhoff orthogonality, J. Inequal. Pure and Appl. Math. 6 (2005), Art. 79. [8] J. Chmieli´ nski, P. W´ojcik, On a ρ-orthogonality, Aequationes Math. 80 (2010), 45–55. [9] J. Chmieli´ nski, P. W´ojcik, ρ-orthogonality and its preservation-revisited, in: Recent Developments in Functional Equations and Inequalities, Banach Center Publications 99 (2013), 17–30. [10] S.S. Dragomir, Semi-Inner Products and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2004. [11] J.R. Giles, Classes of semi–inner–product spaces, Trans. Amer. Math. Soc. 129 (1967), 436–446. [12] P. Grover, Orthogonality to matrix subspaces and a distance formula, Linear Algebra Appl. 445 (2014), 280–288. [13] R.C. James, Orthogonality in normed linear linear spaces, Duke Math. J. 12 (1945), 291–301. [14] R.C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), 265–292.
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[15] C.K. Li, H. Schneider, Orthogonality of matrices, Linear Algebra Appl. 347 (2002), 115–122. [16] G. Lumer, Semi–inner–product spaces, Trans. Amer. Math. Soc. 100 (1961), 29–43. [17] B. Mojˇskerc, A. Turnˇsek, Mappings approximately preserving orthogonality in normed spaces, Nonlinear Anal. 73 (2010), 3821–3831. [18] K. Paul, Translatable radii of an operator in the direction of another operator, Sci. Math. 2 (1999) 119-122. [19] D. Sain, K. Paul, Operator norm attainment and inner product spaces, Linear Algebra Appl. 439 (2013), 2448–2452. [20] D. Sain, K. Paul, S. Hait, Operator norm attainment and Birkhoff-James orthogonality, Linear Algebra Appl. 476 (2015), 85–97. [21] I. Singer, Best approximation in normed linear spaces by elements of linear subspaces, Die Grundlehren der mathematischen Wissenschaften. 171, Springer-Verlag, Berlin-Heidelberg-New York, 1970. [22] P. W´ojcik, Characterization of smooth spaces by approximate orthogonalities, Aequationes Math. 89 (2015), 1189–1194. [23] P. W´ojcik, Orthogonality of compact operators, Expo. Math. 35 (2017), 86–94.
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Jacek Chmieli´nski, Tomasz Stypuła, Paweł Wójcik
PII: DOI: Reference:
S0024-3795(17)30363-4 http://dx.doi.org/10.1016/j.laa.2017.06.001 LAA 14199
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Linear Algebra and its Applications
Received date: Accepted date:
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Approximate orthogonality in normed spaces and its applications Jacek Chmieli´ nski, Tomasz Stypula, Pawel W´ojcik Department of Mathematics, Pedagogical University of Cracow Podchor¸az˙ ych 2, 30-084 Krak´ ow, Poland
Abstract In a normed space we consider an approximate orthogonality relation related to the Birkhoff orthogonality. We give some properties of this relation as well as applications. In particular, we characterize the approximate orthogonality in the class of linear bounded operators on a Hilbert space. Keywords: Birkhoff orthogonality; approximate Birkhoff orthogonality; approximate orthogonality of operators 2010 MSC: 46B20, 46B28, 46C05, 46C50, 47L25, 47L30 1. Introduction In an inner product space (X, ·|·), with the standard orthogonality relation x⊥y ⇔ x|y = 0, a natural way to define an approximate orthogonality (or more precisely ε-orthogonality with ε ∈ [0, 1)) is by: x⊥ε y
⇐⇒
| x|y | ≤ ε x y ,
x, y ∈ X.
It is easy to show, in this setting, the following characterization: x⊥ε y
⇐⇒
∃ z ∈ X : x⊥z, z − y ≤ ε y .
(1.1)
x + y for x = 0 and Indeed, if x⊥ε y, then it is enough to take z = − x|y x2 z = y for x = 0. Conversely, assuming x⊥z and z − y ≤ ε y , we get | x|y | = | x|y − z | ≤ x y − z ≤ ε x y , that is x⊥ε y. Notice that z Email addresses: [email protected] (Jacek Chmieli´ nski), [email protected] (Tomasz Stypula), [email protected] (Pawel W´ ojcik) Preprint submitted to Linear Algebra and Applications
June 2, 2017
was actually chosen from Lin {x, y}. In the subsequent section we will extend the above characterization to the case of Birkhoff orthogonality in a normed space. Then, we will show its usefulness in further considerations connected with the approximate Birkhoff orthogonality, as well as in some applications, in particular, in the operator theory. Throughout the paper we will consider a normed space (X, · ), with dim X ≥ 2, over the scalar field K ∈ {R, C} (which sometimes will be restricted to R). X ∗ denotes the dual space for X; BX and SX will stand for the closed unit ball and the unit sphere, respectively, in X. By ExtA we will mean the set of extremal points of A. 2. Approximate Birkhoff orthogonality 2.1. Birkhoff orthogonality In a normed space X we consider the Birkhoff orthogonality: x⊥B y
⇐⇒
∀λ ∈ K :
x + λy ≥ x
(cf. [6, 13, 14] or a survey [2]). There are numerous characterizations of this relation which refer to various geometrical properties connected with the concept of orthogonality. For x ∈ X let J(x) denote the (nonempty) set of its supporting functionals: J(x) = {ϕ ∈ X ∗ :
ϕ = 1, ϕ(x) = x }.
A well known characterization of the Birkhoff orthogonality is due to James (cf. [14, Corollary 2.2]): x⊥B y
⇐⇒
∃ϕ ∈ J(x) : ϕ(y) = 0.
(2.1)
Another characterization is due to Singer [21]. For a real normed space X and x, y ∈ X we have: x⊥B y
⇐⇒
∃ϕ, ψ ∈ Ext(BX ∗ ) ∩ J(x) ∃α ∈ [0, 1] : αϕ(y) + (1 − α)ψ(y) = 0.
2
(2.2)
2.2. Approximate Birkhoff orthogonality The approximate Birkhoff orthogonality (ε-Birkhoff orthogonality with ε ∈ [0, 1)) can be defined by x⊥εB y
⇐⇒
∀λ ∈ K :
x + λy 2 ≥ x 2 − 2ε x λy
as introduced in [7]. It is easy to see that for ε = 0, ⊥εB = ⊥B and for inner product spaces we have ⊥εB = ⊥ε . Later on, we will give some characterizations of ⊥εB related to (2.1) and (2.2) (Theorems 2.3 and 2.4). We need to recall some related concepts. First are the norm derivatives (cf. [3] for the background and properties of these functionals): ρ± (x, y) = lim± t→0
x + ty 2 − x 2 , 2t
x, y ∈ X, t ∈ R
and related to them orthogonality relations: x⊥ρ± y
ρ± (x, y) = 0.
⇐⇒
The corresponding approximate orthogonalities (defined and investigated in [8, 9]) are: x⊥ερ± y ⇐⇒ |ρ± (x, y)| ≤ ε x y . We recall (cf. [16, 11] or the monograph [10]) that each norm in X admits a semi–inner product (generating this norm), that is a functional [·|·] : X × X → K satisfying the conditions: • [λx + μy|z] = λ [x|z] + μ [y|z] , x, y, z ∈ X, λ, μ ∈ K; • [x|λy] = λ [x|y] , x, y ∈ X, λ ∈ K; • | [x|y] | ≤ x y , x, y ∈ X; • [x|x] = x 2 , x ∈ X. Generally, unless the norm is smooth, there could be many different semi– inner products related to a given norm in X. If the norm in X comes from an inner product, then the inner product itself is the unique semi-inner product. For a fixed semi-inner product [·|·] and x, y ∈ X, we define the semi– orthogonality x⊥s y ⇐⇒ [y|x] = 0 3
and the ε-semi–orthogonality (approximate semi–orthogonality) x⊥εs y
⇐⇒
| [y|x] | ≤ ε x y .
For any x, y ∈ X with x⊥B y there exists a semi-inner product [·|·] such that [y|x] = 0 (cf. [10]). It is also known [9, Theorem 3.2] that if X is a real normed space and ε ∈ [0, 1), then ⊥εs ⊂ ⊥εB , ⊥ερ+ ⊂ ⊥εB and ⊥ερ− ⊂ ⊥εB . Moreover, in a smooth space all the relations coincide, that is ⊥εs = ⊥ερ+ = ⊥ερ− = ⊥εB (cf. also [22]). Let SIP X denote the set of all semi–inner products in X. It is known (cf. [10]) that for a real normed space X
and
ρ+ (x, y) = sup{[y|x] : [·|·] ∈ SIP X }
(2.3)
ρ− (x, y) = inf{[y|x] : [·|·] ∈ SIP X }.
(2.4)
We will also use the following characterization of the ε-Birkhoff orthogonality (cf. [9, Theorem 3.1]): x⊥εB y
⇐⇒
ρ− (x, y) − ε x y ≤ 0 ≤ ρ+ (x, y) + ε x y .
(2.5)
2.3. New characterizations of the approximate Birkhoff orthogonality We start with an auxiliary result. Proposition 2.1. Let X be a real normed space and let x, y ∈ X be fixed, such that x⊥εB y (with some ε ∈ [0, 1)). Then, for each n ∈ N there exists a semi-inner product [·|·]n in X such that 1
x y . (2.6) | [y|x]n | ≤ ε + n Proof. Applying (2.5) and (2.3)-(2.4), for each integer n such that n1 ∈ (0, 1− ε) we may choose semi-inner products [·|·]n and [·|·]n such that 1 1 [y|x]n < ε + (2.7)
x y and − ε +
x y < [y|x]n . n n It follows that for some λn ∈ [0, 1] we have 1 1
x y ≤ λn [y|x]n + (1 − λn ) [y|x]n ≤ ε +
x y . − ε+ n n 4
(We use a simple fact that if a < δ and −δ < b for some a, b ∈ R and δ > 0, then −δ ≤ λa + (1 − λ)b ≤ δ for some λ ∈ [0, 1].) Now, we consider a semi-inner product [·|·]n := λn [·|·]n + (1 − λn ) [·|·]n and from the above inequalities it follows 1 1 − ε+
x y ≤ [y|x]n ≤ ε +
x y . n n
We are in position to prove the main result of this section. Theorem 2.2. Let X be a real normed space. For x, y ∈ X and ε ∈ [0, 1): x⊥εB y
⇐⇒
∃z ∈ Lin {x, y} : x⊥B z, z − y ≤ ε y .
(2.8)
Moreover, the implication “⇐” is true also for complex spaces. Proof. First, suppose that x⊥εB y. We assume that x = 0 (otherwise the result is trivial). It follows from Proposition 2.1 that for an arbitrary n ∈ N there exists a semi–inner product [·|·]n in X such that (2.6) holds. Let ⊥s,n denote the orthogonality relation related to this semi–inner product. We [y|x] define zn := − x2n x + y ∈ Lin {x, y} and it is easy to see that x⊥s,n zn . Since ⊥s,n ⊂ ⊥B , it follows x⊥B zn . Applying (2.6) we estimate zn − y and we get 1
y . (2.9) and
zn − y ≤ ε + x⊥B zn n Notice that zn ≤ 2 y which means that the elements of the sequence (zn )n=1,2,... belong to a closed ball in a two-dimensional space Lin {x, y}. Thus there exists z ∈ Lin {x, y} and a subsequence (znk ) convergent to z. Finally, (2.9) and continuity of the norm yield x⊥B z and z − y ≤ ε y . Now, let X be a real or complex normed space and assume that there exists z ∈ X such that x⊥B z and z − y ≤ ε y . Since x⊥B z, there exists a semi-inner product [·|·] such that [z|x] = 0 and thus | [y|x] | = | [y − z|x] | ≤ y − z x ≤ ε y x , i.e., x⊥εs y. Since ⊥εs ⊂ ⊥εB , we get x⊥εB y.
5
Validity of the implication “⇒” in a complex space remains an open problem. We apply the above result to give another characterization of ⊥εB . Such a characterization was already noticed in [17] for smooth spaces; we will prove it now without this restriction. Theorem 2.3. Let X be a real normed space. For x, y ∈ X and ε ∈ [0, 1): x⊥εB y
⇐⇒
∃ ϕ ∈ J(x) : |ϕ(y)| ≤ ε y .
(2.10)
Moreover, the implication “⇐” is true also for complex spaces. Proof. Let x⊥εB y. Applying (2.8), there exists z ∈ X such that x⊥B z and
z − y ≤ ε y . Due to (2.1) there exists ϕ ∈ J(x) such that ϕ(z) = 0. Therefore, |ϕ(y)| = |ϕ(z) − ϕ(y)| ≤ z − y ≤ ε y , as claimed. For the reverse assume that ϕ ∈ J(x) and |ϕ(y)| ≤ ε y . Fix arbitrarily λ ∈ K. We have then |ϕ(λy)| ≤ ε λy and it follows
x + λy 2 ≥ |ϕ(x + λy)|2 ≥ | |ϕ(x)| − |ϕ(λy)| |2 = |ϕ(x)|2 − 2|ϕ(x)| |ϕ(λy)| + |ϕ(λy)|2 ≥ x 2 − 2ε x λy , which means that x⊥εB y. Proceeding similarly, we generalize Singer’s result (2.2). The following theorem will be useful later (cf. the proof of Theorem 3.6). Theorem 2.4. Let X be a real normed space, let x, y ∈ X, and let ε ∈ [0, 1). Then x⊥εB y
⇐⇒
∃ϕ, ψ ∈ Ext(BX ∗ ) ∩ J(x) ∃α ∈ [0, 1] : |αϕ(y) + (1 − α)ψ(y)| ≤ ε y .
(2.11)
Moreover, the implication “⇐” is true also for complex spaces. Proof. Let x⊥εB y. Applying (2.8), there exists z ∈ X such that x⊥B z and
z − y ≤ ε y and by (2.2) there exist ϕ, ψ ∈ Ext(BX ∗ ) ∩ J(x) and α ∈ [0, 1] such that αϕ(z) + (1 − α)ψ(z) = 0. Therefore, |αϕ(y) + (1 − α)ψ(y)| = |αϕ(y − z) + (1 − α)ψ(y − z)| ≤ α z − y + (1 − α) z − y
= z − y ≤ ε y . 6
For the reverse implication fix arbitrarily λ ∈ K. The functional αϕ + (1 − α)ψ is also in J(x). We have then |(αϕ + (1 − α)ψ)(λy)| ≤ ε λy and similarly as before
x + λy 2 ≥ |(αϕ + (1 − α)ψ)(x + λy)|2 ≥ | |(αϕ + (1 − α)ψ)(x)| − |(αϕ + (1 − α)ψ)(λy)| |2 ≥ x 2 − 2ε x λy , i.e., x⊥εB y. Remark 2.5. Since the (approximate) orthogonality relation in an inner product space is symmetric, we can change the roles of x and y and rewrite (1.1) in an equivalent form: x⊥ε y
⇐⇒
∃ z ∈ X : z⊥y, z − x ≤ ε x .
(2.12)
One may ask whether this characterization remains true in normed spaces with the Birkhoff orthogonality. We show that the answer is negative and actually none of the implications in (2.12) is valid in normed spaces, generally. Example 2.6. Consider the space R2 with the maximum norm
(x1 , x2 ) ∞ := max{|x1 |, |x2 |}. Let x = (1, 1), y = (ε, 1) with ε ∈ (0, 1). Since ρ+ (x, y) = 1 and ρ− (x, y) = ε (cf. [3, Example 2.1.2]) the condition (2.5) yields x⊥εB y follows. Let z be such that z⊥B y. If z ∞ = 1, then z = (1, −1) or z = (−1, 1). Thus it follows that z⊥B y ⇔ z = α(−1, 1). Therefore for each z such that z⊥B y the following inequality holds:
z − x ∞ = max{|1 + α|, |1 − α|} ≥ 1 > ε x ∞ . Now, let x = (1, 1 − ε), y = (1, 0), z = (1 − ε, 1) with ε ∈ (0, 1). Then we have z⊥B y and z − x ∞ ≤ ε x ∞ . On the other hand, ρ− (x, y) = 1, thus ρ− (x, y) − ε x ∞ y ∞ > 0, whence x⊥εB y cannot be true. 3. Applications In this part of the paper we present some applications of the main result from Section 2 in particular normed spaces. 7
3.1. Characterization of approximate orthogonality in L(H) Let L(H) be the space of all linear and bounded operators on a Hilbert space H and let K(H) stand for a subspace of L(H) consisting of all compact operators. For a given T ∈ L(H), we denote MT := {x ∈ SH : T x = T } — the set of points from the unit sphere at which T attains its norm (we follow the notation used in [20, Definition 1.1]). Generally, MT may be empty; however, compactness of T implies that MT = ∅ and this is the case, in particular, if dim H < ∞. ˇ Bhatia and Semrl [5] gave the following characterization of the Birkhoff orthogonality in L(H). Theorem 3.1 ([5], Theorem 1.1, Remark 3.1). Let H be a Hilbert space and let T, S ∈ L(H). Then, the following conditions are equivalent: (i) T ⊥B S; (ii) ∃ (xn )∞ n=1 ⊂ SH : lim T xn = T ,
n→∞
lim T xn |Sxn = 0.
n→∞
Moreover, if dim H < ∞, then each of the above conditions is equivalent to: (iii) ∃ x0 ∈ MT :
T x0 ⊥Sx0 .
Independently, the characterization (i)⇔(ii) was given also by Paul [18, Lemma 2]. Then, the above result was developed by various authors. It is known that the equivalence (i) ⇔ (iii) cannot be extended to L(X) for a finite-dimensional normed space X — a counterexample was given by Li and Schneider [15]. Then Ben´ıtez, Fern´andez and Soriano [4] showed that this equivalence is valid if and only if X is a Hilbert space. Other generalizations or applications of Theorem 3.1 have been obtained, e.g., by Arambaˇsi´c and Raji´c [1], Sain, Paul and Hait [19, 20], Grover [12] and recently by W´ojcik [23]. All these papers, however, dealt with the exact orthogonality of operators. ˇ We will provide some extensions of the result of Bhatia and Semrl by giving a characterization of the approximate orthogonality in L(H). Moreover, in the sequel, we will replace the assumption of finite-dimensionality of H by the compactness of the first operator. Theorem 3.2. Let H be a real Hilbert space, let T, S ∈ L(H) and let ε ∈ [0, 1). Then, the following conditions are equivalent: 8
(1) T ⊥εB S; (2) ∃ (xn )∞ n=1 ⊂ SH : lim T xn = T ,
n→∞
lim | T xn |Sxn | ≤ ε T S .
n→∞
Moreover, if dim H < ∞, then each of the above conditions is equivalent to: (3) ∃ x0 ∈ MT :
| T x0 |Sx0 | ≤ ε T S .
If dim H < ∞ and, additionally, MT ⊂ MS , then each of the above three conditions is equivalent also to: (4) ∃ x0 ∈ MT :
T x0 ⊥ε Sx0 .
Proof. (1) ⇒ (2). Assume T ⊥εB S. It follows from (2.8) that for some U ∈ L(H) we have T ⊥B U and U − S ≤ ε S . Then Theorem 3.1 yields that there exists a sequence (xn ) in SH such that T xn → T and T xn |U xn → 0 as n → ∞. Therefore, | T xn |Sxn | ≤ | T xn |U xn | + | T xn |(S − U )xn | ≤ | T xn |U xn | + T S − U
≤ | T xn |U xn | + ε T S . Without loss of generality the sequence (| T xn |Sxn |)∞ n=1 may be assumed convergent — otherwise we choose an appropriate subsequence (xnk ) of (xn ). Letting n → ∞, we get the desired inequality. (2) ⇒ (1). Assume (2). For an arbitrary scalar λ and each n ∈ N we have
T + λS 2 ≥ T xn + λSxn 2 ≥ T xn 2 + 2λ T xn |Sxn ≥ T xn 2 − 2|λ| | T xn |Sxn |. Letting n → ∞ we get T + λS 2 ≥ T 2 − 2ε T λS , whence T ⊥εB S. Assuming (2) and dim H < ∞, the sequence (xn ) must contain a subsequence convergent to some x0 ∈ SH . Continuity of T and S yields (3). The reverse implication is obvious with xn = x0 for n = 1, 2, . . .. Finally if MT ⊂ MS , then for x0 ∈ MT we have T (x0 ) = T and S(x0 ) = S . Thus (3) is equivalent to (4). 9
Remark 3.3. Notice that implication (4)⇒(1) does not require any additional assumptions. Indeed, assuming (4), for an arbitrary λ we have
T + λS 2 ≥ T x0 + λSx0 2 ≥ T x0 2 − 2ε T x0 λSx0
= T 2 − 2ε T λSx0 ≥ T 2 − 2ε T λS , i.e., T ⊥εB S. However, for (3)⇒(4) the assumption MT ⊂ MS is essential. Indeed, let H be a two-dimensional Hilbert space. Fix ε ∈ [0, 1). Let x, y, z ∈ SH be such that x|y = 2ε and x⊥z. Define T, S ∈ L(H) by: T x := x, T z := 0;
Sx := y, Sz := 2z.
Thus T = 1, T x = 1, S ≥ 2, Sx = 1 and x ∈ MT but x ∈ MS . Since | T x|Sx | = | x|y | = 2ε ≤ ε T S , condition (3) is satisfied. On the other hand | T x|Sx | = | x|y | = 2ε > ε T x Sx , which means that T x ⊥ε Sx (actually T x0 ⊥ε Sx0 for all x0 ∈ MT = {x, −x}). Thus (4) is not satisfied. The condition dim H < ∞ in the above theorem can be replaced by the compactness of the operator T . Theorem 3.4. Let H be a real Hilbert space, let T, S ∈ L(H) and let ε ∈ [0, 1). Assume that MT ⊂ MS and T ∈ K(H). Then T ⊥εB S if and only if ∃ x0 ∈ M T :
T x0 ⊥ε Sx0 .
Proof. Let T ⊥εB S. It follows from (2.8) that for an operator U ∈ L(H) we have T ⊥B U and U − S ≤ ε S . Then from Theorem 3.1 we get a sequence (xn ) in SH satisfying T xn → T and T xn |U xn → 0 as n → ∞. Since BH is weakly compact, there exist a subsequence (xnk ) of (xn ) and x0 ∈ BH such that (xnk ) is weakly convergent to x0 . Since T is compact, we have a strong convergence T xnk → T x0 and it follows T x0 = T . Thus x0 ≥ 1 and hence x0 ∈ SH . Notice that | T xnk |U xnk − T x0 |U x0 | ≤ | T xnk |U xnk − T x0 |U xnk | +| T x0 |U xnk − T x0 |U x0 | ≤ T xnk − T x0 U xnk
+| U ∗ T x0 |xnk − U ∗ T x0 |x0 |. 10
The sequence (U xnk ) is bounded, hence T xnk − T x0 U xnk → 0. Since (xnk ) is weakly convergent to x0 , U ∗ T x0 |xnk → U ∗ T x0 |x0 . The right hand side of the above estimation is convergent to zero, thus we have T xnk |U xnk − T x0 |U x0 → 0. Since T xnk |U xnk → 0, we get T x0 |U x0 = 0. We have x0 ∈ MT ⊂ MS , thus S = Sx0 and
U x0 − Sx0 ≤ U − S ≤ ε S = ε Sx0 . Summarizing, we have obtained T x0 ⊥U x0 and U x0 − Sx0 ≤ ε Sx0 , and therefore, due to (2.8), we get T x0 ⊥ε Sx0 . The reverse implication was proved in Remark 3.3. As a corollary we get the following theorem. Theorem 3.5. Let H be a real Hilbert space. Let T ∈ K(H), S ∈ L(H). Assume that MT ⊂ MS . Then T ⊥εB S
=⇒
S⊥εB T.
(3.1)
Proof. For the proof we need to apply Theorem 3.4, the symmetry of the relation ⊥ε in H and Remark 3.3. Of course, if both operators T, S are compact and MT = MS , then T ⊥εB S
⇐⇒
S⊥εB T.
3.2. Characterization of approximate orthogonality Let K be a locally compact topological space. the space of all real continuous mappings defined supremum norm. We consider a subspace C0 (K) of
in C0 (K) As usual, C(K) denotes on K, enowed with the C(K):
C0 (K) := {f ∈ C(K) : ∀ ε > 0, the set {t ∈ K : |f (t)| ≥ ε} is compact}. For f ∈ C0 (K) the set Mf := {t ∈ K : |f (t)| = f } is always nonempty and compact. Theorem 3.6. Let f, g ∈ C0 (K), f = 0 = g. Assume that Mf is connected. Then, the following conditions are equivalent: 11
(a) f ⊥εB g; (b) ∃ t1 ∈ Mf : |g(t1 )| ≤ ε g . Proof. (a)⇒(b). Let f ⊥εB g. It follows from Theorem 2.4 that there exist extremal points a∗ , b∗ ∈ Ext BC0 (K)∗ and α ∈ [0, 1], such that a∗ (f ) = b∗ (f ) =
f and |αa∗ (g) + (1 − α)b∗ (g)| ≤ ε g . (3.2) It is known (cf. [21, Chapter I, §1, 1.10, 1.11]) that extremal functionals a∗ , b∗ in C0 (K) have the form ±ψu , ±ψw for some u, w ∈ Mf , where ψt : C0 (K) → R is defined as ψt (h) := h(t). Since Mf is connected, f takes on Mf values of the same sign. We can assume without loss of generality that f (t) = f for t ∈ Mf . Then a∗ , b∗ take the form ψu , ψw . Thus a∗ (g) = ψu (g) = g(u) and similarly b∗ (g) = ψw (g) = g(w). Hence from the inequality (3.2) we get |αg(u) + (1 − α)g(w)| ≤ ε g .
(3.3)
If |g(u)| ≤ ε g or |g(w)| ≤ ε g , then we take t1 = u or t1 = w, respectively. If g(u) < −ε g and g(w) > ε g (or g(w) < −ε g and g(u) > ε g ) we use the fact that Mf is connected and the Darboux property for g to get for some t1 ∈ Mf that −ε g ≤ g(t1 ) ≤ ε g as required. We eliminate the remaining cases g(u), g(w) > ε g or g(u), g(w) < −ε g as they would contradict (3.3). (b)⇒(a). Since t1 ∈ Mf , for the functional ϕ = ψt1 (or ϕ = −ψt1 ) we have ϕ = 1 and ϕ(f ) = |f (t1 )| = f , hence ϕ ∈ J(f ). Moreover, |ϕ(g)| = |ψt1 (g)| = |g(t1 )| ≤ ε g and (a) follows from (2.10). 3.3. Characterization of approximate orthogonality in quotient spaces Let M be a closed subspace of a normed space X. Let [x] := x + M be the coset of M with respect to x, and let X/M := {[x] : x ∈ X} be the quotient space endowed with the norm
[x] = dist (x, M ) = inf x − m . m∈M
Notice that x⊥B M ⇐⇒ x = [x] , whence it follows that for any m0 ∈ M we have x − m0 ⊥B M ⇐⇒ x − m0 = [x] . (3.4) 12
We say that M is proximinal if and only if ∀ x ∈ X ∃ y ∈ M : x − y = dist (x, M ). A proximinal subspace is closed. Theorem 3.7. Let X be a real smooth normed space and let M ⊂ X be a proximinal subspace of X. For x, y ∈ X the following conditions are equivalent: (1) [x]⊥εB [y]; (2) ∀ m1 , m2 ∈ M such that x − m1 ⊥B M there is: x − m1 ⊥εB y − m2 ; (3) ∃ m1 , m2 ∈ M such that x−m1 ⊥B M , y −m2 ⊥B M and x−m1 ⊥εB y −m2 . Proof. (1)⇒(2). Assume (1) and fix m1 ∈ M such that x − m1 ⊥B M and m2 ∈ M . It follows from (3.4) and from the definition of ε-orthogonality that for any λ we have
x − m1 2 = [x] 2 ≤ [x] + λ[y] 2 + 2ε [x] λ[y]
≤ x − m1 + λ(y − m2 ) 2 + 2ε x − m1 λ(y − m2 ) , which means x − m1 ⊥εB y − m2 . The implication (2)⇒(3) is obvious (here we use proximinality of M ). Now, we prove (3)⇒(1). Let m1 , m2 ∈ M be such that x−m1 ⊥B M , y −m2 ⊥B M and x−m1 ⊥εB y −m2 . It follows from Theorem 2.2 that there exists z ∈ Lin {x − m1 , y − m2 } such that x − m1 ⊥B z and
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