On a ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}-Orthogonally Additive Mappings

We show that a real normed linear space endowed with the ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}-orthogonality relation, in general need not be an orthogonality space in the sense of Rätz. However, we prove that ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}-orthogonally additive mappings defined on some classical Banach spaces have to be additive. Moreover, additivity (and approximate additivity) under the condition of an approximate orthogonality is considered.


Introduction
Inner product spaces are by all means the most natural venue for orthogonality, allowing the definition: x⊥y ⇔ x|y = 0. However, analogous relations may be considered also in normed linear spaces, as well as in more general settings.
In a real normed linear space (X, · ), for two vectors x, y ∈ X, one can consider for example the Birkhoff-James orthogonality ⊥ B (see [2,15]) defined by or the isosceles orthogonality ⊥ i (see [15]) defined by or many others. Moreover, some axiomatic definitions of the orthogonality in linear spaces (or even more general structures) are known, among them the one formulated by Rätz [18] (compare with [12]). A real linear space X of dimension The research of Pawe l Wójcik was partially supported by National Science Centre, Poland under Grant Miniatura 2, No. 2018/02/X/ST1/00313. at least 2 with a binary relation ⊥ ⊆ X × X is called an orthogonality space (in the sense of Rätz) whenever the following conditions are satisfied: (ort1) x⊥0 and 0⊥x for all x ∈ X; (ort2) if x, y ∈ X \ {0} and x⊥y, then x and y are linearly independent; (ort3) if x, y ∈ X and x⊥y, then αx⊥βy for all α, β ∈ R; (ort4) for any two-dimensional subspace P of X, for every x ∈ P and for every λ ∈ [0, +∞) there exists y ∈ P such that x⊥y and x + y⊥λx − y. It is easily seen that any inner product space is an orthogonality space in the above sense. Not so easily, but still the same can be shown for an arbitrary real normed linear space with the Birkhoff-James orthogonality ⊥ B (see [18]).
Yet another axiomatic definition of orthogonality, considered merely on groups, was introduced by Fechner and Sikorska [10]; we refer to it in Sect. 4.
Let X be a suitable algebraic structure with an orthogonality relation ⊥ (in whatsoever sense) and let G be a group (usually abelian). The following conditional functional equation, with f : X → G, will be the key notion of the paper: The function f is then called an orthogonally additive mapping and the history of that concept goes back nearly a century. We refer to surveys by Rätz [19] and more recent by Sikorska [20] for motivations, history, various aspects and problems connected with the subject.
In the basic case of X being a real inner product space it is known (cf. [5,18,26]) that any orthogonally additive mapping f must be of the form with unique additive mappings a : R → G and b : X → G. This yields, in particular, that an orthogonally additive mapping defined on an inner product space need not be additive. Notice that commutativity of the group G need not be assumed here (see the recent proof by Toborg [26,Theorem 3.3]). In more general cases, orthogonally additive mappings have been also widely investigated (see e.g., [6,12,18]). We recall here the theorem of Rätz, Baron and Volkmann concerning the orthogonal additivity for mappings defined on an orthogonality space.
if and only if there exist an additive mapping ϕ : X → G and a biadditive and symmetric mapping Φ: X × X → G such that and Φ(x, y) = 0 for all x, y ∈ X such that x⊥y. It follows that all Birkhoff-James orthogonally additive mappings (⊥ Badditive mapings), i.e., such that x⊥ B y implies f (x + y) = f (x) + f (y), have the form (1.3).
Quite surprisingly, it turns out that in the setting of the orthogonality space (X, ⊥), all orthogonally additive mappings are unconditionally additive, unless X is an inner product space, i.e., unless there exists an inner product ·|· in X such that x⊥y if and only if x|y = 0. Equivalently, only inner product spaces admit nonzero and even orthogonally additive mappings. This was proved by Szabó [23] when dim X ≥ 3 and by Yang [27] for the remaining case dim X = 2. Therefore, all ⊥ B -additive mappings are additive, unless the norm · comes from an inner product. Actually, for this particular orthogonality, the phenomenon was observed earlier -see [16,21,22]. The above property, however, is not restricted to orthogonality spaces. In particular, in a normed linear space of the dimension greater than 2 and with the isosceles orthogonality ⊥ i , the existence of orthogonally additive mappings which are not additive also characterizes inner product spaces -see [24,25].
Let us go back to the case of an inner product space X. For a nonnegative constant ε, a notion of ε-orthogonality (approximate orthogonality) of vectors x, y ∈ X can be introduced in a natural way by Surely, ⊥ 0 = ⊥ and since for ε ≥ 1 the Cauchy-Schwarz inequality leads to ⊥ ε = X 2 , we restrict the range of ε to the interval [0, 1).
Having defined an approximate orthogonality on X, we may consider, for a group G and a mapping f : X → G, a stronger than (1.1) condition x,y∈ X. (1.4) Obviously, for ε = 0, (1.4) coincides with (1.1).
Although an orthogonally additive mapping defined on an inner product space need not be additive, any solution of (1.4) with a strictly positive ε is unconditionally additive. Theorem 1.2. Let X be a real inner product space with dim X ≥ 2, let G be a group and let ε ∈ (0, 1). A function f : X → G satisfies (1.4) if and only if f is additive.
Let us stress that we do not assume commutativity of G here, although, as in the rest of the paper, we use an additive notation.
Proof. It is clear that additivity of f implies (1.4). For the reverse, assume (1.4) whence f must be of the form (1.2). In order to prove additivity of f it is enough to show that the mapping a vanishes. Of course a(0) = 0, so we fix an arbitrary t ∈ R \ {0} and by proving that a(t) = 0 we will finish the proof. Let us choose two unit vectors u, v such that u|v = ε. Then for x := t 2ε u and y := v we have consequently a( x + y 2 ) = a( x − y 2 ) and a(t) = a(2 x|y ) = 0 follows.
The above considerations and results have motivated our work, which is devoted to an orthogonality relation ⊥ ρ defined in a real normed linear space and to the corresponding with it orthogonal additivity.
The paper is organized as follows. In Sect. 2, using the notion of norm derivatives, we introduce the mapping ρ and define the ρ-orthogonality. In the subsequent part we consider ρ-orthogonally additive mappings defined on some particular real normed linear spaces. In Sect. 4 we extend Theorem 1.2 to normed linear spaces and we investigate mappings which are approximately additive under the condition of an approximate orthogonality. We finish our paper with some concluding remarks and open problems.

Norm Derivatives and ρ-Orthogonality
In this section we define and consider yet another orthogonality relation in a real normed linear space (X, · ). First, we recall the notion of the so-called (right and left) norm derivatives ρ + , ρ − : X × X → R (see e.g., [1,7,9]): Convexity of the norm yields that the limits exist and the above definitions are meaningful. It is also natural to consider (cf. [17]) the mapping ρ : X ×X → R being the arithmetic mean of ρ + and ρ − , i.e., The following properties of ρ will be usefull (for the proofs consult, e.g., [1,9,17]): Vol. 75 (2020) On a ρ-Orthogonally Additive Mappings Page 5 of 17 108 Note, that ρ need not be continuous with respect to the first variable. The mapping ρ can thus be regarded as a substitute of an inner product in X (and so can be ρ + and ρ − ). The following definition is therefore natural and it plays a crucial role in the present paper. We define a ρ-orthogonality relation, denoted by ⊥ ρ , as It is easy to check that the relation ⊥ ρ satisfies the Rätz axioms (ort1), (ort2) and (ort3). However, as shown in the example below, the axiom (ort4) need not be fulfilled in general. We are aware that this contradicts the statement of [1, Proposition 2.8.1], the proof of which is unfortunately incorrect.
In this particular space, explicit formulas for ρ + and ρ − can be obtained (see [ Consider the vector (1, 1 4 ) ∈ R 2 ; it follows from (2.1) that for any (α, β) ∈ R 2 there is The space l 2 ∞ is not exceptional in lacking the (ort4) condition; on the contrary, the family of such spaces is quite large. Let (X 1 , · 1 ), (X 2 , · 2 ) be nontrivial real normed linear spaces. By X 1 ⊕ ∞ X 2 we denote the product space X 1 × X 2 with the norm (x 1 , x 2 ) ∞ := max{ x 1 1 , x 2 2 }. Now, the space X 1 ⊕ ∞ X 2 contains a subspace isometric to l 2 ∞ and therefore, as a consequence of what we observed in Example 2.1, it cannot satisfy (ort4).
Notice that some classical spaces like l ∞ , c, c o have the form X 1 ⊕ ∞ X 2 (for example c o is isometric to R ⊕ ∞ c o ). The fact that the space (X, ⊥ ρ ) need not be an orthogonality space in the sense of Rätz, motivates our investigations in the next section.

ρ-Orthogonal Additivity
We aim at a characterization of ρ-orthogonally additive mappings, i.e., those satisfying x⊥ ρ y =⇒ f (x + y) = f (x) + f (y) (3.1) in spaces for which (in view of Theorem 2.2) we cannot apply Theorem 1.1.
In the first part of this section, we consider the case where the domain is a product of two real normed linear spaces and the relation ⊥ ρ corresponds to the relevant norm · ∞ in that space. We start with the following lemma. Lemma 3.1. Let (X 1 , · 1 ) and (X 2 , · 2 ) be real normed linear spaces that yield the product space X 1 ⊕ ∞ X 2 and the orthogonality relation ⊥ ρ in it. For arbitrary x 1 ∈ X 1 and x 2 ∈ X 2 the following statements are true.
In the following theorem, given two normed linear spaces, we characterize ρ-orthogonally additive mappings defined on the product of these spaces and taking values in an arbitrary group. We stress here that the commutativity of the target group is not assumed.

Theorem 3.2.
Let G be a group, let (X 1 , · 1 ) and (X 2 , · 2 ) be nontrivial real normed linear spaces and let f : Then f is additive.
Since ⊥ ρ ⊆ ⊥ B (see e.g., [7,8]), from Theorem 3.2 we get immediately a novel, in a sense, property of Birkhoff-James orthogonally additive mappings. Theorem 3.3. Let G be a group, let X 1 and X 2 be nontrivial real normed linear spaces and let f : Then f is additive.
We emphasize that commutativity of G is not assumed, so the above result cannot be derived from the already known ones. In particular, it does not follow from Theorem 1.1 or from the results of Szabó [23] and Yang [27], even though (X 1 ⊕ ∞ X 2 , ⊥ B ) is an orthogonality space.
Let us consider now a classical Banach space C[0, 1] of continuous realvalued functions on [0, 1], with the supremum norm. For mappings ϕ, ψ ∈ C[0, 1] defined by the subspace span{ϕ, ψ} ⊆ C[0, 1] is isometric to l 2 ∞ . Thus the orthogonality ⊥ ρ in C[0, 1] does not satisfy (ort4). Nevertheless, we are able to characterize ρ-orthogonally additive mappings defined on that space. In the first result we make no commutativity assumption for the target group.

Theorem 3.4. Let G be a group and let
Then there exist additive mappings a, b : Proof. For a fixed number t 1 ∈ (0, 1) let M t1 , N t1 ⊆ C[0, 1] be the subspaces defined by It is easy to check that ker η t1 = M t1 ⊕N t1 . Moreover, it is not difficult to show that the subspace M t1 ⊕N t1 ⊆ C[0, 1] is isometrically isomorphic to the Banach space M t1 ⊕ ∞ N t1 . Thus, we may identify M t1 ⊕ ∞ N t1 with ker η t1 and we may consider M t1 ⊕ ∞ N t1 as a closed subspace of C[0, 1] with codim M t1 ⊕ ∞ N t1 = 1. From Theorem 3.2 we know that f | ker ηt 1 is additive. Now, fix a number t 2 ∈ (0, 1) \ {t 1 } and define η t2 : . Similarly as before one proves that f | ker ηt 2 is additive.
Assuming that G is abelian, the sum of mappings a and b is additive, whence our result takes the following form. A natural problem arises. Problem 3.6. Is the commutativity of G a necessary assumption in Theorem 3.5?
In order to answer this question positively, it suffices to prove that if two mappings a, b : C[0, 1] → G are additive and their sum f := a + b satisfies (3.3), then f is additive. We can relate this problem with the result of Toborg [26,Corollary 3.4] who proved that in the case of an inner product space X in the domain, the subgroup of G generated by the image f (X) of an orthogonally additive mapping f is abelian.
Another question is whether the results obtained for X 1 ⊕ ∞ X 2 and C[0, 1] can be extended to other, or perhaps all, spaces. The exact statement of this problem concludes the section. Problem 3.7. Is it true that in any normed linear space X which is not an inner product space, any ρ-orthogonally additive mapping is additive?

Approximate ρ-Orthogonal Additivity and Stability
In the final part of the paper we deal with approximate solutions of equation (3.1) as well as with its stability.
We start with a simple general observation. Let (F, +) be a semigroup, let (G, +) be an abelian group, and let Δ ⊆ F × F and U ⊆ G be nonempty subsets. Suppose that mappings f, g : x ∈ F and g is additive on Δ, i.e., Then for a pair (x, y) ∈ Δ we have −g(x + y) + g(x) + g(y) = 0 and also Adding it all (with the help of commutativity), we obtain Example 4.1. Keeping the above notation, suppose that a mapping g : F → G satisfies the property (4.1). For each element x ∈ F , using the axiom of choice, we take an arbitrary a x ∈ g(x) + U and define a function f : ∈ U for each x ∈ F and it follows that f satisfies (4.2). This shows that the family of mappings satisfying (4.2) can be large and these mappings may by very irregular.
Assume now that for a binary relation ⊥ in F we have x⊥y ⇔ (x, y) ∈ Δ and assume that the target group G is replaced by a normed linear space (Y, · ) with U being a closed ball centred at zero, with the radius δ ≥ 0. Suppose that a mapping g : F → Y satisfies (4.1), which now takes the form x⊥y =⇒ g(x + y) = g(x) + g(y), x,y∈ F.
If a mapping f : F → Y satisfies f (x) − g(x) ≤ δ for all x ∈ F , then f satisfies (4.2), the meaning of which is The last property will be of our interest in the present section with the role of ⊥ played by the ρ-orthogonality or an approximate ρ-orthogonality. We would like to ask a reverse, in a sense, question whether an approximately orthogonally additive mapping can be approximated by an orthogonally additive (or additive) one. The first result in this direction was obtained by Ger and Sikorska in [11] and then improved by Fechner and Sikorska in [10]. In the latter paper the authors considered an abelian group A with a binary relation ⊥ on it, with the properties: (ort-a) if x, y ∈ A and x⊥y, then x⊥ − y, −x⊥y and 2x⊥2y; (ort-b) for every x ∈ A there exists y ∈ A such that x⊥y and x + y⊥x − y. With such settings, (A, ⊥) is a generalization of an orthogonality space in the sense of Rätz. It can be derived from the main theorem of [10] that for A being a uniquely 2-divisible abelian group, Y being a Banach space and f : A → Y satisfying, with δ ≥ 0, the condition there exists a mapping g : A → Y such that x⊥y =⇒ g(x + y) = g(x) + g(y), x,y∈ A and f (z) − g(z) ≤ 5δ for all z ∈ A. For a pair of vectors x, y in a real normed linear space X we define their approximate ρ-orthogonality, or to be more precise, ε-ρ-orthogonality, with ε ∈ [0, 1), by x⊥ ε ρ y ⇐⇒ |ρ (x, y)| ≤ ε x y (see [7,8]). If the norm comes from an inner product, then ⊥ ε ρ coincides with ⊥ ε defined in the introduction.
As we have actually observed in Example 2.1, the relation ⊥ ρ does not satisfy (ort-b) in l 2 ∞ . It can be shown that the relation ⊥ ε ρ , for ε < 1 4 , does not satify (ort-b) in that space as well (we omit, however, a tedious verification of this fact). Now, we are ready to prove the main results of this section. We will call a mapping f satisfying (4.4), D-additive.