A unitary operator A unitary operator U acting on a Hilbert space whose spectrum is C = {z Î C; |z| = 1 }.

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If H is a separable infinite dimensional Hilbert space with an orthogonal basis (xn)n Î Z, we define Uxn = xn+1. Then U is isometric and surjective, so it is a unitary operator. By Lemma 3.2.13 of [K&R1], sp(U) Í C. If l Î C and xn = (2n+1)[(-1)/2] åk = -nn l-k xk, then ||xn || = 1 and ||(U - lI)xn|| = (2n+1)[(-1)/2]||åk = -nn l-k xk+1 - åk = -nn l-(k-1) xk || = (2n+1)[(-1)/2]||l-n xn+1 - ln+1 x-n || = 2[1/2](2n+1)[(-1)/2] ® 0.
Therefore by the same lemma, l Î sp(U). Thus sp(U) = C.

Ref.
[K&R1] R.V. Kadison and J.R. Ringrose, Fundamental of the theory of Operator Algebras (I), Acad. Press, 1983.

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On 22 Feb 2001, 00:21.