Self-Adjoint Operators
Definition
An operator $T \in \mathcal{L} (V)$ is called
self-adjoint (or
Hermitian) if the adjoint
of $T$ is itself.
In symbols, $T$ is self-adjoint if $T = T^*$.
In other words, $T$ is self-adjoint if and
only if
\[
\ip{Tv, w} = \ip{v, Tw} \quad \text{for all } v, w \in V
\]
Properties
Suppose $S, T \in \mathcal{L} (V)$ are
self-adjoint.
The $S + T$ are self-adjoint.
Also $\lambda T$ is adjoint for all real
$\lambda $.
Notation
We will see that the adjoint on $\mathcal{L} (V)$ plays a role similar to complex conjugation on $\C $. The self-adjoint operators will seen to be analogous to the real numbers. A complex number is real if and only if $z = \Cconj{z}$. Similarly, an operator is self-adjoint if and only if $T = T^*$.
Characterization for complex space
Suppose $V$ is a complex inner product space
and let $T \in \mathcal{L} (V)$.
Then
\[ T = T^* \quad \iff \quad (\forall v \in V)\;(\ip{Tv, v} \in \R ) \]