As we know, the spectrum of an operator $T$ has a standard decomposition into three parts:
- a point spectrum, consisting of eigenvalues of $T$ ;
- a continuous spectrum, consisting of the scalars that are not eigenvalues but make the range of $T-\lambda$ a proper dense subset of the space;
- a residual spectrum, consisting of all other scalars in the spectrum.
My question is are there any spectrum has continuous spectrum but no residual spectrum? Or conversely, a spectrum has residual spectrum but no continuous spectrum?