In Functional calculus it has written that the spectrum of a compact operator is real. Is it true or it is a misprint?
2026-03-27 01:45:55.1774575955
Is the Spectrum of a compact operator real?
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The standing assumption in this section is that the operator is self-adjoint. In this case the spectrum is indeed real (and compactness is not needed for that). If the operator is not self-adjoint, then this is no longer true. Any matrix with non-real eigenvalues gives you a counterexample.