On ring derivations and quadratic functionals

Summary In this note the general form of ring derivations on a function algebra is obtained. In particular, we give a very simple proof of the following result of Nandakumar: a ring derivation on a function algebra is trivial provided that the Choquet boundary of the algebra contains a dense sequentially non-isolated set. Using this theorem we prove a result concerning representability of quadratic functionals by bilinear forms.This work was supported by the Research Council of Slovenia.     

On quadratic forms

The present paper is related to the Jordan—von Neumann characterization of inner product spaces, to the Halperin problem concerning quadratic forms, to some results of the present author on quadratic and sesquilinear forms and to recently obtained results of C. T. Ng and of J. Vukman.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.     

On orthogonally additive mappings,IV

Summary 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.Making a further step in the modifications of the axioms for the space (X, +, ), the additive/quadratic representation of the solutions of (*) now can be proved in a much more general situation which includes also the case of the orthogonality induced by an isotropic symmetric bilinear functional.     

A characterization of Gaussian law in Hilbert space

Summary In the present paper the main result is the following:Let be a real separable Hilbert space. LetX andY be two independently distributed random variables taking values in . Then the random variablesX+Y andX–Y are independently distributed if and only if each ofX andY follows a Gaussian law.The proof of the above result depends on the solution of a functional equation in the general framework of a real separable Hilbert space.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth     

A characterization of generalized inner product spaces (ϕ-orthogonally additive mappings,III)

Summary The conditional Cauchy functional equation for a mappingF: (X, +, ) (Y, +), i.e.,F(x + y) = F(x) + F(y) for allx, y X withx y, (*) on a real vector space equipped with an abstract relation (calledorthogonality), was first studied by Gudder and Strawther in 1975. They defined by a system consisting of five axioms and described the general hemi-continuous real valued solution of (*) showing that the existence of non-trivial even ones characterize inner product orthogonality. Using the more restrictive axioms of Rätz (introduced in 1980 to obtain the general solution without regularity conditions: odd solutions are additive, while the even ones are quadratic), recently we have proved the same assuming arbitrary mappingsF with values in an abelian group but for dimX 3. In 1989, Rätz and the author modified the system of axioms so that it should include the orthogonality induced by an isotropic symmetric bilinear form and still ensure the additive/quadratic representation.In this context, the main purpose of this note is to characterize on a real vector space the symmetric bilinear orthogonality as the essentially unique extension of an orthogonality relation satisfying certain weak axioms and admitting non-trivial even hemi-continuous solutions of (*) with values in a Hausdorff topological abelian group.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth     

Almost summing mappings

We introduce a general definition of almost p-summing mappings and give several concrete examples of such mappings. Some known results are considerably generalized and we present various situations in which the space of almost p-summing multilinear mappings coincides with the whole space of continuous multilinear mappings. Received: 17 June 2002     

Summability and estimates for polynomials and multilinear mappings

In this paper we extend and generalize several known estimates for homogeneous polynomials and multilinear mappings on Banach spaces. Applying the theory of absolutely summing nonlinear mappings, we prove that estimates which are known for mappings on ?_p spaces in fact hold true for mappings on arbitrary Banach spaces.     

Christensen measurable solutions of generalized Cauchy functional equations

Aequationes mathematicae -     

Jordan *-derivation pairs and quadratic functionals on modules over *-rings

Summary We give a new simple proof of Šemrl’s recent representation theorem for quasi-quadratic functions acting on unital modules and then show that our approach also gives a certain extension of Šemrl’s result. This paper is intended to point out the usefulness of the ternary point of view even when we are dealing with problems which involve only binary structure. Dedicated to the memory of Professor Gy?rgy Szabó Supported in part by the Slovene Ministry of Science grant P1-5505-0101/93.     

Presemiclosed mappings

The purpose of this paper is to investigate the properties of presemiclosed mappings and improve some already established results. Moreover, it is shown that, if the domain space isT ₀, then optimal continuity and homeomorphism coincide (Theorem 4.18).     

On mappings preserving equilateral triangles

Let E be a Euclidean space, dim E 2. We say that f : E E preserves equilateral triangles if for all triples of points x, y, z E we have We show that if E is a finite-dimensional Euclidean space, dim E 2, f:E E is measurable and preserves equilateral triangles, then it is a similarity transformation (an isometry multiplied by a positive constant). Moreover, in spaces at least three-dimensional we get a similarity transformation without any regularity assumption. Some generalizations as well as some interesting examples are also presented in the paper.     

Some remarks on derivations in Banach algebras and related results

Summary In the first section of this paper we consider some functional equations which are closely connected to derivations (i.e. additive mappings with the propertyD(ab) = aD(b) + D(a)b) on Banach algebras. IfD is a derivation on some algebraA, then the equationD(a) = – aD(a ^–1 )a holds for all invertible elementsa A. It seems natural to ask whether this functional equation characterizes derivations among all additive mappings. It is too much to expect an affirmative answer to this question in arbitrary algebras, since it may happen that even in normed algebras the group of all invertible elements contains only scalar multiples of the identity. We try to answer the question above in Banach algebras, since in Banach algebras invertible elements exist in abundance. In the second section of the paper we prove some results concerning representability of quadratic forms by bilinear forms.     

Pythagorean orthogonality and additive mappings

Summary In this note we consider a real normed vector spaceX equipped with the isosceles orthogonality or the Pythagorean orthogonality, both of them defined by R. C. James. It is known that any odd, isosceles orthogonally additive mapping fromX into an Abelian group is unconditionally additive whenever dimX 3. Also, it is worth mentioning that this result was the first of this sort based on a non-homogeneous relation. In this context, we derive here the same for the other non-homogeneous orthogonality, the Pythagorean one, answering in part a pretty old and famous question. The proof uses the corresponding result for isosceles orthogonality and a detailed analysis of the geometry of normed spaces.Dedicated to Professor Jürg Rätz on the occasion of his 60th birthday     

On bounded time-dependent perturbations of operator cosine functions

In this note it is shown that the Cauchy problem associated with the infinitesimal generatorA of a strongly continuous operator cosine function remains uniformly well-posed under bounded time-dependent perturbations ofA.