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The spectrum of a unitary operator is non-empty

Let X be a (possibly infinite-dimensional!) Hilbert space, and let U:X→XU: X \to X be a unitary operator. I need to show that the spectrum σ(U)={λ∈C:U−λ id is not invertible}\sigma(U) = \{ \lambda \in \mathbb{C} : U - \lambda \, \mathrm{id} \text{ is not invertible} \} of U is non-empty. The hint suggests that I can do this by considering T=i(U+id)(U−id)−1T = i(U + \mathrm{id})(U - \mathrm{id})^{-1}. 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. U=(i00i)U = \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix} has T=(1001)T = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}. Is there a way to do it using the hint, or is there a better way to do it without the hint?
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csmith101
Original post by Zhen Lin
Let X be a (possibly infinite-dimensional!) Hilbert space, and let U:X→XU: X \to X be a unitary operator. I need to show that the spectrum σ(U)={λ∈C:U−λ id is not invertible}\sigma(U) = \{ \lambda \in \mathbb{C} : U - \lambda \, \mathrm{id} \text{ is not invertible} \} of U is non-empty. The hint suggests that I can do this by considering T=i(U+id)(U−id)−1T = i(U + \mathrm{id})(U - \mathrm{id})^{-1}. 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. U=(i00i)U = \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix} has T=(1001)T = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}. 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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