Positive operator

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

In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator A acting on an inner product space is called positive-semidefinite (or non-negative) if, for every x \in \operatorname{Dom}(A) , \langle Ax, x\rangle \in \mathbb{R} and \langle Ax, x\rangle \geq 0 , where \operatorname{Dom}(A) is the domain of A . Positive-semidefinite operators are denoted as A\ge 0 . The operator is said to be positive-definite, and written A>0 , if \langle Ax,x\rangle>0 for all x\in\mathop{\mathrm{Dom}}(A) \setminus \{0\} . Many authors define a positive operator 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. Take the inner product \langle \cdot, \cdot \rangle to be anti-linear on the first argument and linear on the second and suppose that A is positive and symmetric, the latter meaning that \langle Ax,y \rangle= \langle x,Ay \rangle . Then the non negativity of \begin{align} \langle A(\lambda x+\mu y),\lambda x+\mu y \rangle =|\lambda|^2 \langle Ax,x \rangle + \lambda^* \mu \langle Ax,y \rangle+ \lambda \mu^* \langle Ay,x \rangle + |\mu|^2 \langle Ay,y \rangle \\[1mm] = |\lambda|^2 \langle Ax,x \rangle + \lambda^* \mu \langle Ax,y \rangle+ \lambda \mu^* (\langle Ax,y \rangle)^* + |\mu|^2 \langle Ay,y \rangle \end{align} for all complex \lambda and \mu shows that \left|\langle Ax,y\rangle \right|^2 \leq \langle Ax,x\rangle \langle Ay,y\rangle. It follows that \mathop{\text{Im}}A \perp \mathop{\text{Ker}}A. If A is defined everywhere, and \langle Ax,x\rangle = 0, then Ax = 0. \langle Ax,y\rangle = \frac{1}{4}({} & \langle A(x+y),x+y\rangle - \langle A(x-y),x-y\rangle \\[1mm] & {} - i\langle A(x+iy),x+iy\rangle + i\langle A(x-iy),x-iy\rangle) \end{align} and the fact that \langle Ax,x\rangle = \langle x,Ax\rangle, for positive operators, show that \langle Ax,y\rangle = \langle x,Ay\rangle, so A is symmetric. In contrast with the complex case, a positive-semidefinite operator on a real Hilbert space H_\mathbb{R} may not be symmetric. As a counterexample, define A : \mathbb{R}^2 \to \mathbb{R}^2 to be an operator of rotation by an acute angle \varphi \in ( -\pi/2,\pi/2). Then \langle Ax,x \rangle = \|Ax\|\|x\|\cos\varphi > 0, but A^* = A^{-1} \neq A, so A is not symmetric. The symmetry of A implies that \operatorname{Dom}A \subseteq \operatorname{Dom}A^* and A = A^*|_{\operatorname{Dom}(A)}. For A to be self-adjoint, it is necessary that \operatorname{Dom}A = \operatorname{Dom}A^*. In our case, the equality of domains holds because H_\mathbb{C} = \operatorname{Dom}A \subseteq \operatorname{Dom}A^*, so A is indeed self-adjoint. The fact that A is bounded now follows from the Hellinger–Toeplitz theorem. This property does not hold on H_\mathbb{R}. The definition of a quantum system includes a complex separable Hilbert space H_\mathbb{C} and a set \cal S of positive trace-class operators \rho on H_\mathbb{C} for which \mathop{\text{Trace}}\rho = 1. The set \cal S is the set of states. Every \rho \in {\cal S} is called a state or a density operator. For \psi \in H_\mathbb{C}, where \|\psi\| = 1, the operator P_\psi of projection onto the span of \psi is called a pure state. (Since each pure state is identifiable with a unit vector \psi \in H_\mathbb{C}, some sources define pure states to be unit elements from H_\mathbb{C}). States that are not pure are called mixed.

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