The Physical Meaning of Projectors in Quantum Mechanics

A projector is an observable - you can directly check that it is Hermitian $|L\rangle\langle L|^\dagger = |L\rangle \langle L|$. As to interpretation - a projector onto a single state will measure the value $1$ for definite if the system is in that state. If the system is in an orthogonal state it will measure $0$. Therefore you can think of projectors as operators whose measurement corresponds to asking a binary question. Any measurement you can think of can be approximated by a series of binary questions and so its not surprising that any observable can be decomposed into such projectors.

As for your second question: I don't see why not. The notation $L^2$ is confusing though - I'd stick to calling this $L_1+L_2$ or similar. Note that this operator is not a projector. It's still Hermitian, and it's a reasonable thing to consider if you have two subsystems on which $L$ is itself sensible to consider.

This is only answering your question 1.

Like all Hermitian operators, the operator $P_L=|L\rangle\langle L|$ represents a physical observable. It is easy to verify that this operator has the two eigenvalues:

• $1$, with eigenstate $|L\rangle$
• $0$, with eigenstate $|R\rangle$

So the corresponding physical observable is a Boolean observable, for the property "the system is in state $|L\rangle$". The measurement result will be either true (1) or false (0). And after this measurement the system will be in state $|L\rangle$ or $|R\rangle$, respectively.