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# The spectrum of a unitary operator is non-empty Watch

1. Let X be a (possibly infinite-dimensional!) Hilbert space, and let be a unitary operator. I need to show that the spectrum of U is non-empty. The hint suggests that I can do this by considering . I can see that T is self-adjoint, and if I can further show that T is compact, then I can apply a sledgehammer (namely, the spectral theorem for compact self-adjoint operators) to get the result, but it seems that T is not compact in general. If I can show T is not invertible that also shows the result, but T can be invertible - e.g. has . Is there a way to do it using the hint, or is there a better way to do it without the hint?
2. (Original post by Zhen Lin)
Let X be a (possibly infinite-dimensional!) Hilbert space, and let be a unitary operator. I need to show that the spectrum of U is non-empty. The hint suggests that I can do this by considering . I can see that T is self-adjoint, and if I can further show that T is compact, then I can apply a sledgehammer (namely, the spectral theorem for compact self-adjoint operators) to get the result, but it seems that T is not compact in general. If I can show T is not invertible that also shows the result, but T can be invertible - e.g. has . Is there a way to do it using the hint, or is there a better way to do it without the hint?
I've actually come across a very similar problem myself. Out of interest, did trying to use compactness work out? Or did you find a different way to use the hint? (or neither...)

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Updated: December 29, 2010
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