Euclidean Jordan Algebra (3) - Linear Transformations

For any element $a\in V$, the Lyapunov transformation $L_a:V\to V$ and the quadratic representation $P_a:V\to V$ are defined as

$$\begin{array}{rl}{L}_a(x)& :=a\circ x,\\ P_a(x)& :=(2L_a^2-L_{a^2})(x)=2a\circ (a\circ x)-a^2\circ x,\end{array}$$ for all $x\in V$. These transformations are linear and self-adjoint on $V$. In the following example, we describe these transformations in the Euclidean Jordan algebras defined in the above.

Example 4. 

1. For ${\mathbb{R}}^n$, the above transformations are given by $L_a(x)=a\ast x$ and $P_a(x)=a^2\ast x$. Here, note that $x\ast y$, a component-wise product, can be regarded as $\mathrm{Diag}(x)y$ where $\mathrm{Diag}(x)$ denotes a diagonal matrix of size $n$ whose diagonal entries are the entries of $x$.
2. For ${\mathcal{S}}^n$, the above transformations are given by $L_A(X)=\frac{1}{2}(AX+XA)$ and $P_A(X)=AXA$.
3. For ${\mathcal{L}}^n$, the above transformations are
   $$L_a(x)=\left(\begin{array}{cc}{a}_1& {\overline{a}}^{\mathrm{\top }}\\ \overline{a}& a_{1}I\end{array}\right)\left(\begin{array}{c}{x}_1\\ \overline{x}\end{array}\right),P_a(x)=\left(\begin{array}{cc}\Vert a{\Vert }^2& 2a_1{\overline{a}}^{\mathrm{\top }}\\ 2a_1\overline{a}& (a_1^2-\Vert \overline{a}{\Vert }^2)I+2\overline{a}{\overline{a}}^{\mathrm{\top }}\end{array}\right)\left(\begin{array}{c}{x}_1\\ \overline{x}\end{array}\right).$$

In any Euclidean Jordan algebra $V$, one con define automorphism groups in the following way:

Let $\mathrm{\Lambda }:V\to V$ be a linear transformation. Then, $\mathrm{\Lambda }$ is an algebra automorphism if it is invertible and $\mathrm{\Lambda }(x\circ y)=\mathrm{\Lambda }(x)\circ \mathrm{\Lambda }(y)$ for all $x,y\in V$. The set of all algebra automorphisms of $V$ is denoted by $\mathrm{Aut}(V)$. 

A linear transformation $\mathrm{\Gamma }:V\to V$ is a (symmetric) cone automorphism if $\mathrm{\Gamma }(V_{+})=V_{+}$. Such automorphism is automatically invertible. The set of all cone automorphisms of $V$ is denoted by $\mathrm{Aut}(V_{+})$. Is is immediate that $\mathrm{Aut}(V)\subseteq \mathrm{Aut}(V_{+})$. Moreover, if $\mathrm{\Gamma }\in \mathrm{Aut}(V_{+})$, then so are ${\mathrm{\Lambda }}^{-1}$ and ${\mathrm{\Lambda }}^{\mathrm{\top }}$.

Example 5. 

1. For ${\mathbb{R}}^n$, it is easily seen that $\mathrm{Aut}({\mathbb{R}}^n)$ consists of permutation matrices, and any element in $\mathrm{Aut}({\mathbb{R}}_{+}^n)$ has a form $DP$, where $P$ is a permutation matrix and $D$ is a diagonal matrix with positive diagonal entries.
2. For ${\mathcal{S}}^n$, it is known that corresponding to any $\mathrm{\Lambda }\in \mathrm{Aut}({\mathcal{S}}^n)$, there exists an orthogonal matrix $U\in {\mathbb{R}}^{n\times n}$ such that $\mathrm{\Lambda }(X)=UX{U}^{\mathrm{\top }}$ for all $X\in {\mathcal{S}}^n$. Also, for $\mathrm{\Gamma }\in \mathrm{Aut}({\mathcal{S}}_{+}^n)$, there exists an invertible matrix $Q\in {\mathbb{R}}^{n\times n}$ such that $\mathrm{\Gamma }(X)=QX{Q}^{\mathrm{\top }}$ for all $X\in {\mathcal{S}}^n$.
3. For ${\mathcal{L}}^n$, it is known that any algebra automorphism $\mathrm{\Lambda }$ can be written as $\mathrm{\Lambda }=\left(\begin{array}{cc}1& 0\\ 0& U\end{array}\right)$, where $U$ is an $(n-1)\times (n-1)$ orthogonal matrix. The explicit description of $\mathrm{Aut}({\mathcal{L}}_{+}^n)$ is not known, yet. However, $\mathrm{\Gamma }\in \mathrm{Aut}({\mathcal{L}}_{+}^n)$ if and only if there exists $\mu >0$ such that ${\mathrm{\Gamma }}^{\mathrm{\top }}J\mathrm{\Gamma }=\mu J$, where $J=\mathrm{diag}(1,-1,\dots ,-1)\in {\mathbb{R}}^{n\times n}$. 

A linear transformation $\mathrm{\Phi }:V\to V$ is said to be orthogonal if $⟨\mathrm{\Phi }(x),\mathrm{\Phi }(y)⟩=⟨x,y⟩$ for all $x,y\in V$. The set of all orthogonal linear transformations is denoted by $\mathrm{Orth}(V)$.

A linear transformation $\mathrm{\Phi }:V\to V$ is is doubly stochastic if $\mathrm{\Phi }$ is positive (i.e., $\mathrm{\Phi }(V_{+})\subseteq V_{+}$), unital (i.e., $\mathrm{\Phi }(e)=e$), and trace preserving (i.e., $\mathrm{tr}(\mathrm{\Phi }(x))=\mathrm{\Phi }(x)$ for all $x\in V$). We denote the set of all doubly stochastic linear transformations by $\mathrm{DS}(V)$.

For a Euclidean Jordan algebra $V$, the followings hold:

• The positivity of $\mathrm{\Phi }:V\to V$ is equivalent to that of ${\mathrm{\Phi }}^{\mathrm{\top }}:V\to V$.
• When $V$ carries a canonical inner product, the trace preserving (unital) property of $\mathrm{\Phi }$ is equivalent to the unital  (trace preserving) property of ${\mathrm{\Phi }}^{\mathrm{\top }}$. In particular, $\mathrm{\Phi }$ is doubly stochastic if and only if its transpose is doubly stochastic.
• When $V$ carries a canonical inner product, it is known that $\mathrm{Aut}(V)=\mathrm{Aut}(V_{+})\cap \mathrm{Orth}(V)$. Furthermore, if $V$ is simple, we have $\mathrm{Aut}(V)=\mathrm{Aut}(V_{+})\cap \mathrm{DS}(V)$.