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 $T = T\adj$ .