In this talk, a matrix group is a closed subgroup of the group of (bounded) invertible linear transformations on a finite-dimensional real inner product space.
A semi-FTvN system is a triple $(V, W, λ)$, where $V$ and $W$ are real inner product
spaces and $λ : V → W$ is a map satisfying the conditions $||λ(x)|| = ||x||$ and
$⟨x, y⟩ ≤ ⟨λ(x), λ(y)⟩$ for all $x, y ∈ V$ . In such a system, we consider the automorphism group consisting of invertible linear transformations $A$ on $V$ satisfying
the condition $λ(Ax) = λ(x)$ for all $x ∈ V$ . Examples of semi-FTvN systems
include FTvN systems (such as Euclidean Jordan algebras and systems induced
by complete isometric hyperbolic polynomials). We will show that complete
hyperbolic polynomials give rise to semi-FTvN systems.
Given a matrix group $G$ on a finite-dimensional real inner product space $V$
(such as the automorphism group of a semi-FTvN system) and $a, b ∈ V$ , we
say that a commutes with b relative to $G$ if $⟨Da, b⟩ = 0$ for all $D$ in $Lie(G) :=
{L ∈ L(V ) : exp(tL) ∈ G, ∀ t ∈ R}$. In this talk, we describe some examples
and show how this commutativity relation appears as a necessary optimality
condition in certain optimization problems.
