Compact Operator On Hilbert Space - Unitary Operator

Unitary Operator

The spectrum of a unitary operator U lies on the unit circle in the complex plane; it could be the entire unit circle. However, if U is identity plus a compact perturbation, U has only countable spectrum, containing 1 and possibly, a finite set or a sequence tending to 1 on the unit circle. More precisely, suppose U = I + C where C is compact. The equations UU* = U*U = I and C = U − I show that C is normal. The spectrum of C contains 0, and possibly, a finite set or a sequence tending to 0. Since U = I + C, the spectrum of U is obtained by shifting the spectrum of C by 1.