dynamical operator and $SU(n+1)$

Question

I want to know precisely by example

1. what is dynamical operator?

2. what is the relationship between dynamical operators and the $SU(n+1)$

3. How to show all the eigen states of a dynamical operator form a complete set of basis vectors of a Hilbert Space?(Any quantum state of a quantum system can be regarded as a vector in the Hilbert space)