Abstract

In this paper we consider the generalization of the orthogonality equation. Let S be a semigroup, and let H, X be abelian groups. For two given biadditive functions A:S^2rightarrow X, B:H^2rightarrow X and for two unknown mappings f,g:Srightarrow H the functional equation B(f(x),g(y))=A(x,y)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} B(f(x),g(y))=A(x,y) \\end{aligned}$$\\end{document}will be solved under quite natural assumptions. This extends the well-known characterization of the linear isometry.

Highlights

  • The above equation was generalized in normed spaces X, Y by considering a norm derivative ρ+(x, y):= x x+ty − x t instead of inner product, i.e

  • Another generalization of the orthogonality equation in Hilbert spaces H, K is to look for the solutions of f (x)|g(y) = x|y, x, y ∈ H, (2)

  • In this paper we will give a natural generalization of such functional equations in the case of abelian groups

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Summary

Introduction

It is easy to check that, if f : H → K satisfies f (x)|f (y) = x|y , f is an linear isometry. The above equation was generalized in normed spaces X, Y by considering a norm derivative ρ+(x, y):= x. X+ty − x t instead of inner product, i.e. with an unknown function f : X → Y. Note that if the norm comes from an inner product ·, · , we obtain ρ+(x, y) = x|y. Another generalization of the orthogonality equation in Hilbert spaces H, K is to look for the solutions of f (x)|g(y) = x|y , x, y ∈ H,

Results
Preliminaries
Main results
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Applications
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