Unitary Operator - Properties

Properties

  • The spectrum of a unitary operator U lies on the unit circle. That is, for any complex number λ in the spectrum, one has |λ|=1. This can be seen as a consequence of the spectral theorem for normal operators. By the theorem, U is unitarily equivalent to multiplication by a Borel-measurable f on L²(μ), for some finite measure space (X, μ). Now U U* = I implies |f(x)|² = 1 μ-a.e. This shows that the essential range of f, therefore the spectrum of U, lies on the unit circle.

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