30.2 Self-Adjoint Elements

Definition 30.2.1 (Self-Adjoint).label Let $A$ be an involutive algebra over $\complex$ and $x \in A$, then $x$ is self-adjoint if $x = x^{*}$. The space $A_{sa}= \bracs{x \in A| x = x^*}$ is the self-adjoint part of $A$, and:

1. (1)

   $A_{sa}$ is a $\real$ subspace of $A$.

2. (2)

   $A = \complex(A_{sa})$ as a vector space.

3. (3)

   For each $x \in A$, let

   \[\text{Re}(x) = \frac{x + x^{*}}{2}\quad \text{Im}(x) = \frac{x - x^{*}}{2i}\]

   
   

   then $\text{Re}(x), \text{Im}(x) \in A_{sa}^{2}$ and $x = \text{Re}(x) + i\text{Im}(x)$.

4. (4)

   For each $x \in A$, $x^{*} = \text{Re}(x) - i\text{Im}(x)$.

Definition 30.2.2 (Normal).label Let $A$ be an involutive algebra over $\complex$ and $x \in A$, then the following are equivalent:

1. (1)

   $\text{Re}(x)\text{Im}(x) = \text{Im}(x)\text{Re}(x)$.

2. (2)

   $x^{*}x = xx^{*}$.

If the above holds, then $x$ is normal.