What are the steps in order to determine the projection valued measure associated with a possibly unbounded self adjoint operator $A$?
Is it straight forward to apply the recipe ? Or, does each operator has his own tricks?
Examples of operators : parity operator, angular momentum operator...
Thank you for your help.
Introductory courses are somewhat skewed towards nice, computable examples, but in general, you don't have a chance to determine the projection-valued measure explicitly. For a compact operator or an operator with compact resolvent, this task boils down to finding all eigenvalues and associated eigenvectors, and this is already too hard. Keep in mind that the eigenvalue equation can be about any homogeneous linear equation and one should not expect to have a unified solution theory for such a broad class of equations.
Often, techniques from partial differential equations come in handy (for differential or integral operators) and sometimes you can use Fourier analysis as in the case of constant-coefficient differential operators on $\mathbb{R}^d$, and it is of course almost trivial for multiplication operators. This is more or less all I regularly use.