Let $A:X\rightarrow Y$, Where $X,Y$ are Hilbert Spaces, be bounded linear operator. Then show that, for every $\alpha \gt 0$, $A^*A+\alpha I$ is bounded below. Where $A^*=$ Adjoint of $A$.
Attempt: An operator$A$ is called bounded below if there exist $c\gt 0$ such that $$||Ax||\ge c||x||, \forall x\in X$$
$||(A^*A+\alpha I)x||=||A^*Ax+\alpha x||\le(||A^*A||+\alpha)||x||$ I'm stuck here. Help is needed. Thanx in advance.