Stability of the Generalized Orthogonality Functional Equation

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:

• (A) ||T(x)|| =||x||.

• (B) T(x) = 0 if and only if x = 0.

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

• (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.

• (E) T(µx) = ±µT(x).