Advertisement

Stability of the Generalized Orthogonality Functional Equation

  • Donald H. Hyers
  • George Isac
  • Themistocles M. Rassias
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 34)

Abstract

Let (E,.) denote a real Hilbert space of dimension greater than one, and let T: E → E be a mapping which satisfies the functional equation
$$\left| {T\left( x \right) \cdot T\left( y \right)} \right| = \left| {x \cdot y} \right|$$
(9.1)
known as the generalized orthogonality equation. We will follow the work of C. Alsina and J.L. Garcia-Roig (1991) (see also Th.M. Rassias (1997)). It is easy to see that any solution of (9.1) will satisfy the following conditions for x, y in E and real µ:
  1. (A)

    ||T(x)|| =||x||.

     
  2. (B)

    T(x) = 0 if and only if x = 0.

     
  3. (C)

    T(x) • T(y) = 0 if and only if xy = 0.

     
  4. (D)

    I cos A(x, y) I = I cos A(T (x), T (y))1, where A(u, v) denotes the angle between the vectors u and v.

     
  5. (E)

    T(µx) = ±µT(x).

     

Keywords

Hilbert Space Functional Equation Real Hilbert Space Complex Hilbert Space Linear Isometry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Donald H. Hyers
  • George Isac
    • 1
  • Themistocles M. Rassias
    • 2
  1. 1.Department of Mathematics and Computer ScienceRoyal Military College of CanadaKingstonCanada
  2. 2.Department of MathematicsNational Technical University of AthensAthensGreece

Personalised recommendations