Operator

A linear operator $T \in \mathcal L(V)$ on the Hilbert space $V$ acts on a vector to produce another vector $T : V \to V$ .

An operator can be defined in terms of its action on a basis. For a finite dimensional Hilbert space, an operator can be written as a matrix. By linearity

\begin{align*} T \ket{v_j} = \sum_i T_{ij} \ket{v_i} \end{align*}

where $T_{ij}$ are the matrix elements.

We define the hermitian conjugate or adjoint $T\adj$ of an operator $T$ by the relation

\begin{align*} \braket{a|Tb} = \braket{T\adj a|b}. \end{align*}

An operator $T$ is hermitian if $T = T\adj$ . It is anti-hermitian if $T = -T\adj$ .

$T$ is unitary if $T\adj = T\inv$ . A unitary operator is norm-preserving. The complex exponential of any hermitian or anti-hermitian operator $A$ is unitary $T = e^{iA}$ .

$T$ is normal if $[T,T\adj] = 0$ . Unitary operators and hermitian operators are normal.