An observable is given by $$\sum\limits_{n= 1}^N a_n|a_n\rangle\langle a_n | $$ Here $\langle a_n |a_m\rangle = \delta_{nm}$. What are the possible measurement results corresponding to the operator A and what are the possible states of the system after measurement?
How do I calculate the eigenvalues and states when only the operator is given? If I proceed with the answer as follows: $$A \vert a_n \rangle = \sum\limits_{n= 1}^N a_n |a_n\rangle \langle a_n|a_n\rangle = \sum\limits_{n= 1}^N a_n |a_n\rangle= a_n \vert a_n \rangle $$ Can I say that $a_n$ is the eigenvalue and $|a_n\rangle$ the eigenstate?
If $A$ is observable (operator), we have: $$A \vert a_m \rangle = \sum\limits_{n= 1}^N a_n |a_n\rangle \langle a_n|a_m\rangle = \sum\limits_{n= 1}^N a_n |a_n\rangle \delta_{mn} = a_m \vert a_m \rangle $$
then yes, $a_n$ is eigenvalue and $|a_n\rangle$ is its corresponding eigenstate (eigenvector).
Also, elements of A can be found as:
$$ A_{nm} = \langle a_n|A|a_m\rangle = \langle a_n|(a_m \vert a_m \rangle)= a_m \delta_{mn} $$