Multiplicative Cauchy functional equation on symmetric cones

Abstract

Abstract. We solve the logarithmic Cauchy functional equation in the symmetric cone with respect to two different multiplication algorithms. We impose no regularity assumptions on respective functions.Mathematics Subject Classification. Primary 39B52.

The generalized fundamental equation of information on symmetric cones

“…Without any regularity assumptions, it was solved on the Lorentz cone in [18]. Recently, the general form of w 1 -logarithmic functions, without any regularity assumptions, was given in [9]. In this case f (x) = H(det x), where H is a generalized logarithmic function, i.e.,…”

“…Note that due to (8) the function H(det x) is always a solution to (9), regardless of the choice of the multiplication algorithm w, but may be not the only one -the best example is the multiplication algorithm related to the triangular group (see [9,Theorem 3.5] and comment before Corollary 3.9). If a w-logarithmic functions f is additionally K-invariant (f (x) = f (kx) for any k ∈ K), then H(det x) is the only possible solution (Theorem 3.3).…”

“…Let f : Ω → R be a function satisfying (9). Assume additionally that f is K-invariant, i.e., f (kx) = f (x) for any k ∈ K and x ∈ Ω.…”

“…Δ r (x) sr . Then log Δ s (x) for some s ∈ R r is the general form of a continuous w 2 -logarithmic function (see [9,Theorem 3.5]) and we have the following Observe that κ log det x is the only continuous w 1 -and w 2 -logarithmic function, therefore we obtain the following f (x) = κ 1 log det(e − x) + κ 2 log det x + log Δ s3 (I − x) + C 1 , g(x) = κ 1 log det(e − x) + log Δ s3 (x) + C 2 , h(x) = κ 1 log det(e − x) + κ 2 log det(I − x) + log Δ s3 (x) + C 3 , k(x) = κ 1 log det(e − x) + κ 2 log det x + C 4 , and C 1 + C 2 = C 3 + C 4 .…”

“…The general form of w-logarithmic Cauchy functions for two basic examples of multiplication algorithms w 1 (x) = x 1/2 and w 1 (x) = t x without any regularity assumptions was recently found in [9]. Later on we will write w(x)y = w(x)·y·w T (x) and, in this case, w(x) denotes the linear operator acting on Ω + .…”