Stability of the Generalized Orthogonality Functional Equation
Chapter
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
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:
$$\left| {T\left( x \right) \cdot T\left( y \right)} \right| = \left| {x \cdot y} \right|$$
(9.1)
- (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).
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.
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© Springer Science+Business Media New York 1998