Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 106 / 5 / 3

The hypothesis is possible only for $t\ge 0$. For every $\lambda \in \sigma (x)\subset \mathbb{R}$,so $0\le \lambda \le 2t$. The spectral characterization of a positive element in a C*-algebra gives $x\ge 0$.