Self-Adjoint Operators

Definition

An operator $T \in L (V)$ 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

$⟨ T v, w ⟩ = ⟨ v, T w ⟩ for all v, w \in V$

Properties

Suppose

S, T \in L (V)

are self-adjoint. The

S + T

are self-adjoint. Also

λ T

is adjoint for all real

λ

Notation

We will see that the adjoint on $L (V)$ 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 $z = z^{*}$ . 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 L (V)

. Then

$T = T^{*} ⟺ (\forall v \in V) (⟨ T v, v ⟩ \in R)$