The hypothesis is possible only for $t\geq0$. For every $\lambda\in\sigma(x)\subset\mathbb R$,
$$
|t-\lambda|\leq\|t1-x\|\leq t,
$$
so $0\leq\lambda\leq2t$. The <spectral characterization of a positive element in a C-star algebra>[spectral characterization of a positive element in a C*-algebra] gives $x\geq0$.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-106/5/3/solution