I'm studying the maths of quantum mechanics with Schuller's lectures and Simon Rea's lecture notes and defining PVM they say:
Definition: A map P: $\sigma(\mathcal{O}_\mathbb{R}) \to \mathcal{L}(H)$ is called a projection-valued measure if it
satisfies the following properties.
- $\forall \Omega \in \sigma(\mathcal{O}_\mathbb{R}): P(\Omega)^* = P(\Omega)$
- $\forall \Omega \in \sigma(\mathcal{O}_\mathbb{R}): P(\Omega)\circ P(\Omega) = P(\Omega)$
- $P(\mathbb{R}) = id_H$
- For any pairwise disjoint sequence {$\Omega_n$}$_{n \in \mathbb{N}}$ in $\sigma(\mathcal{O}_\mathbb{R})$ and any $\psi \in H$,
$$\sum^\infty_{n=0}P(\Omega_n)\psi = P\bigg(\bigcup^\infty_{n=0}\Omega_n\bigg)\psi$$
Following that, they prove a series of properties such:
$$\text{if } \Omega_1 \subseteq \Omega_2, \text{ then } ran(P(\Omega_1)) \subseteq ran(P(\Omega_2))$$
To prove that they say that $P(\Omega)\geq 0 $, but I can't see why/how these previous 4 properties that define a PVM, imply such positiveness of the $P(\Omega)$ operator. Can someone help me? Such property is what I need to prove some other result.
Best Answer
For the sake of closing the question:
As stated in the comments, $P(\Omega)$ is an orthogonal projection for all $\Omega\in\sigma(\mathcal{O}_{\mathbb{R}}).$ In particular, if $x\in H,$ then $$\langle P(\Omega) x,x\rangle=\langle (P(\Omega))^2 x,x\rangle=\langle P(\Omega) x,(P(\Omega))^*x\rangle=\langle P(\Omega)x,P(\Omega)x\rangle=\|P(\Omega) x\|^2,$$ where we have used the first two properties in your definition (i.e. the orthogonal projection properties).