Jürg Rätz

On orthogonally additive mappings

The conditional Cauchy functional equationF: (X, +, ⊥) → (Y, +), F(x + y) = F(x) + F(y) x, y ∈ X, x ⊥ y, has first been studied under regularity (mainly continuity and boundedness) conditions and by referring to the inner product and the Birkhoff—James orthogonalities (A. Pinsker 1938, K. Sundaresan 1972, S. Gudder and D. Strawther 1975). The latter authors proposed an axiomatic framework for the space (X, +, ⊥), and it then became possible to modify their axioms so that it could be proved without any regularity condition that the odd solutions of (*) are additive and the even ones are quadratic (cf., e.g., ([8], [12]). The results obtained included the classical case of the inner product orthogonality as well as the three following generalizations thereof: (i) Birkhoff—James orthogonality on a normed space, (ii) orthogonality induced by a non-isotropic sesquilinear functional, (iii) semi-inner product orthogonality.

On Approximately Additive Mappings

The stability question for additive mappings under various conditions on their domains and ranges is studied. The main aspects are existence, uniqueness, and continuity of an approximating additive mapping (Sections 4 and 5). Suitable examples demonstrate the limits of the scope of our theorems (Section 6). The monogenic subsets of the domain and the behavior of the mappings on these turn out to be of central importance.

On Wigner's theorem: Remarks, complements, comments, and corollaries

SummaryIn this paper we present a unified treatment of Wigners unitarity-antiunitarity theorem simultaneously in the real and the complex case. Its elementary nature, emphasized by V. Bargmann in 1964, is underlined here by removing unnecessary hypotheses, the most important being the completeness of the inner product spaces involved. At the end, we shall obtain connections to some recent results in geometry.

Orthogonality and additivity modulo a subgroup

SummaryLetE be a real inner product space of dimension at least 2,F a topological Abelian group, andK a discrete subgroup ofF. Assume also thatF is continuously divisible by 2 (that is, the functionu → 2u is a homeomorphism ofF ontoF). Iff: E → F fulfils the conditionf(x + y) − f(x) − f(y) ∈ K for all orthogonalx, y ∈ E and is continuous at the origin then there exist continuous additive functionsa: R → F andA: E → F such thatf(x) − a(∥x∥2)− A(x) ∈ K for everyx ∈ E.

Convexity of power functions with respect to symmetric homogeneous means

It is well known that the power function \( x \mapsto {x^p} \) on \( \mathbb{R}_{ + }^{*} \) is strictly Jensen-convex if p 2−p>0, strictly Jensen-concave if p 2−p<0, Jensen-convex and Jensen-concave if p 2−p=0. These Jensen type properties are based upon the arithmetic mean A : \( A:\mathbb{R}_{ + }^{*} \times \mathbb{R}_{ + }^{*} \to \mathbb{R}_{ + }^{*} \). It is the purpose of this paper to investigate the convexity/concavity classification of the power functions for symmetric homogeneous means on \( \mathbb{R}_{ + }^{*} \) other than A. In Section 3, a convexity/concavity criterion is presented, and in Section 4 this is applied to the families of Stolarsky means and Gini means (both containing A) as well as to weighted geometric means.

Zur Linearität verallgemeinerter Modulisometrien

In this paper we study the functional equations (I) and (II) (cf. the introduction) for mappingsϕ from anR-moduleM into anR-moduleN; R is a ring with an identity and with an antiautomorphism, andf:M × M → R, g:N × N →R are bilinear (or somewhat more general) mappings. Theorems 3 and 7 — the main results of the paper — provide sufficient conditions which guarantee thatϕ is linear. These theorems generalize the authors previous results in [5].

Cauchy functional equation problems concerning orthogonality

Summary. We confine ourselves to the conditional Cauchy functional equation of orthogonal additivity and deal with problems which arise in connection with the following subjects: Dimension 2 versus higher dimensions; Dependence on the range space; Modules as domain spaces; Gleasons Theorem; Disjoint additivity on normed Riesz spaces.

An extremal problem related to probability

SummaryWe consider a walk from a stateA1 to a stateAn+1 in which the probability of remaining atAi ispi, and the probability of progressing fromAi toAi+1 is 1 −pi. The probabilityWnk of reachingAn+1 fromA1 in exactlyn + k steps can then be expressed as a polynomial of degreen + k in then variablesp1,⋯,pn. We determine the maximum value ofWnk and the (unique) choice (p1,⋯,pn) for which this extremum occurs.

On the homogeneity of additive mappings

LetK be a ring with an identity 1 ≠ 0 andM, L two unitaryK-modules. Then, for any additive mappingf:M →L, the setHf:={α ∈ K ∣ f(αx)=αf(x) for allx ∈ M} forms a subring ofK, the ‘homogeneity ring’ off. It is shown that, forM ≠ {0},L ≠ {0} and any subringS ofK for whichM is a freeS-module, there exists an additive mappingf:M→L such thatHf=S. This result is applied to the four Cauchy functional equations, and it leads also to an answer to the question as to whether it is possible to introduce onM a multiplication ·:M × M → M makingM into a ring but not into aK-algebra.

Convex functions with respect to an arbitrary mean

For a mean M, a notion of M-convex function is introduced. A general criterion for the M-convexity of the sum of M-convex functions is given. As an application, we present conditions under which polynomials and the exponential functions are convex with respect to some of the Stolarsky means.