User:Caliburn/s/fa/1

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Theorem

Let $\struct {\HH, \innerprod \cdot \cdot_\HH}$ be a Hilbert space.

Let $A : \HH \to \HH$ be a bounded linear operator.

Let $A^* : \HH \to \HH$ be the adjoint of $A$.

Let $\map \sigma A$ be the spectrum of $A$.

Let $\map \sigma {A^*}$ be the spectrum of $A^*$.


Then:

$\lambda \in \map \sigma A$ if and only if $\overline \lambda \in \map \sigma {A^*}$