Abstract from the Pacific Journal of Mathematics

Volume 181, Number 2, December 1997

Title: On Orthogonally Exponential Functionals

Author: Janusz Brzdek

Abstract:

Let (X, \perp) be an orthogonality space and g:X -> C, g(X) not equal to {0}, be an orthogonally exponential functional, hemicontinuous at the origin. We show that then one of the follwing two conditions is valid:

(i) There are unique linear functionals a_1, a_2:X -> R with

g(x)=\exp(a_1(x)+ia_2(x)) for x in X

(ii) there are a \perp-equivalent inner product (.,.) in X, c in C and unique linear functionals a_1, a_2:X -> R such that

g(x)=\exp(a_1(x)+ia_2(x) +c||x||^2) for x in X

where ||x||= (x,x) for x in X.

We also prove some auxiliary results concerning functions f mapping a real linear (orthogonality) space X into a commutative group (G,+) and satisfying one of the following two conditions:

f(x+y)+f(x-y)- 2f(x)-2f(y) in K for x, y in X,
f(x+y)-f(x)-f(y) in K whenever x\perp y,

where K is a subgroup of G.

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