Definition 2

Let E be a complex Hilbert space and let $u$ be an endomorphism of E. The numerical image of $u$ is the set of complex numbers of the form $\langle x|u(x)\rangle$, where $x$ ranges over the unit sphere of E. We denote by $\iota (u)$ the numerical image of $u$.

The numerical image of $u^*$ is the image of $\iota (u)$ under complex conjugation. For any complex numbers $\lambda$ and $\mu$, the numerical image of $\lambda u+\mu1_E$ is equal to $\lambda \iota (u) +\mu$.