Exercise 10, § 7

Let $E$ be a complex Hilbert space. Let $x_1$ and $x_2$ be endomorphisms of E. now prove that the spectrum of the endomorphism $x_1\otimes x_2$ of $E\widehat{\otimes}_2E$ is equal to Sp($x_1$) Sp($x_2$). (Identify $\overline{E}\widehat{\otimes}_2E$ with the space of Hilbert–Schmidt operators from E into E, cf. EVT 5, p. 52, th. 1, b), and apply the preceding exercise.)