Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 326 / 2 / b / i / Solution

Let $T_{\alpha }=A^{\ast }A+\alpha I$. For every $f\in X$,Thus $T_{\alpha }$ is a coercive operator. If it is invertible and $T_{\alpha }f=g$, the lower bound and the Cauchy-Schwarz inequality giveand therefore $\Vert T_{\alpha }^{-1}g\Vert \le {\alpha }^{-1}\Vert g\Vert$. Hence