Positive operator (Hilbert space)

In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator ${\displaystyle A}$ acting on an inner product space is called positive-semidefinite (or non-negative) if, for every ${\displaystyle x\in \mathop {\text{Dom}} (A)}$, ${\displaystyle \langle Ax,x\rangle \in \mathbb {R} }$ and ${\displaystyle \langle Ax,x\rangle \geq 0}$, where ${\displaystyle \mathop {\text{Dom}} (A)}$ is the domain of ${\displaystyle A}$. Positive-semidefinite operators are denoted as ${\displaystyle A\geq 0}$. The operator is said to be positive-definite, and written ${\displaystyle A>0}$, if ${\displaystyle \langle Ax,x\rangle >0,}$ for all ${\displaystyle x\in \mathop {\mathrm {Dom} } (A)\setminus \{0\}}$.

Many authors define a positive operator ${\displaystyle A}$ to be a self-adjoint (or at least symmetric) non-negative operator. We show below that for a complex Hilbert space the self adjointness follows automatically from non-negativity. For a real Hilbert space non-negativity does not imply self adjointness.

In physics (specifically quantum mechanics), such operators represent quantum states, via the density matrix formalism.