This generalises Sz .-Nagy's dilation theorem as all contractions have the unit disc as a spectral set.
2.
A consequence of this is that any operator with a simply connected spectral set " X " has a minimal normal " ?X " dilation.
3.
It can be shown that if an operator T has X as a spectral set, and \ mathcal { R } ( X ) is a Dirichlet algebra, then T has a normal boundary dilation.
4.
More generally, if \ mathcal { R } ( X ) is a Dirichlet algebra, any operator " T " with X as a spectral set will have a normal \ partial X dilation with this property.
5.
To see that this generalises Sz .-Nagy's theorem, note that contraction operators have the unit disc "'D "'as a spectral set, and that normal operators with spectrum in the unit circle " ? " "'D "'are unitary.
मोबाइल