This is another exercise from Golan's book.
Problem: Let V be an inner product space over C and let α be an endomorphism of V satisfying α∗=−α, where α∗ denotes the adjoint. Show that every eigenvalue of α is purely imaginary.
My proposed solution is below.
Answer
Let me show another argument which applies to a more general setting: if α is a linear operator on a Hilbert space satisfying α∗=−α, then the spectrum of α is purely imaginary (i.e. real part equal zero).
Indeed, one simply needs to notice that α−λid is invertible if and only if (α−λid)∗ is invertible. As (α−λid)∗=α∗−¯λid=−α−¯λid=−(α+¯λid), we conclude that any λ in the spectrum of α satisfies ¯λ=−λ.
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