= Solution
The equation $(A^*A+\alpha_*I)u_*=A^*f$ is the <Tikhonov normal equation>. For $u=u_*+h$, expand the <Tikhonov regularization> functional:
$$
\begin{aligned}
\phi_{\alpha_*}(u)
&=\phi_{\alpha_*}(u_*)
+2\operatorname{Re}\langle A^*(Au_*-f)+\alpha_*u_*,h\rangle_X\\
&\quad+\|Ah\|_Y^2+\alpha_*\|h\|_X^2.
\end{aligned}
$$
The normal equation makes the linear term zero. The final two terms are nonnegative and are strictly positive for $h\ne0$ because $\alpha_*>0$. Thus $u_*$ is the unique global minimizer.
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