how to calculate spectral measure for a given normal operator for example right shift operator?
There is an example in
MR1483073 (98g:46001) Meise, Reinhold ; Vogt, Dietmar . Introduction to functional analysis. Translated from the German by M. S. Ramanujan and revised by the authors. Oxford Graduate Texts in Mathematics, 2. The Clarendon Press, Oxford University Press, New York, 1997. x+437 pp. ISBN:
The spectral decomposition for the unbd. operator $-i d/dx$ can be computed from that of a multiplication operator via the Fourier-Plancherel transform. That's in Kato's book on functional analysis.
Another way is to compute first of all the polynomial calculus for your operator $T$, which sometimes is doable and then make a guess about how $f(T)$ looks like for say continuous functions f (of course you must then find a proof and show that if polynomials $p_n$ tend to $f$, then $p_n(T)$ goes to $f(t)$. If you send me an email, I can send you some examples.