= Solution
For a model with $k$ fitted parameters, the <Akaike information criterion> is
$$
\operatorname{AIC}=-2\ell(\widehat\theta)+2k,
$$
where $\ell$ is the log-likelihood evaluated at the maximum-likelihood estimate.
In the normal linear model with known $\sigma^2$,
$$
-2\ell(\beta)
=n\log(2\pi\sigma^2)+\frac1{\sigma^2}\|Y-X\beta\|^2.
$$
The maximum-likelihood estimator is the ordinary least-squares estimator, with fitted mean $\widehat\mu=HY$, where
$$
H=X(X^TX)^{-1}X^T.
$$
Since only the $p$ components of $\beta$ are fitted,
$$
\operatorname{AIC}
=n\log(2\pi\sigma^2)
+\frac1{\sigma^2}\|Y-\widehat\mu\|^2+2p.
$$
After multiplying by $\sigma^2$ and discarding the model-independent constant, this is exactly <Mallows Cp>:
$$
\boxed{C_p=\|Y-\widehat\mu\|^2+2p\sigma^2}.
$$
To compare it with prediction error, write $Y=\mu+\varepsilon$ and $Y^*=\mu+\varepsilon^*$, where $\mu=X\beta$, the two errors are independent, and both have covariance $\sigma^2I$. Since $H\mu=\mu$, $H$ is an orthogonal projection of rank $p$, and
$$
Y^*-\widehat\mu=\varepsilon^*-H\varepsilon.
$$
The cross term has zero expectation, so
$$
\operatorname{MSPE}
=\mathbb E\|\varepsilon^*\|^2
+\mathbb E\|H\varepsilon\|^2
=n\sigma^2+p\sigma^2.
$$
On the other hand,
$$
\mathbb E\|Y-\widehat\mu\|^2
=\mathbb E\|(I-H)\varepsilon\|^2
=\sigma^2\operatorname{tr}(I-H)
=(n-p)\sigma^2.
$$
Therefore the <unbiased prediction-error identity for ordinary least squares> gives
$$
\mathbb E(C_p)=(n-p)\sigma^2+2p\sigma^2
=(n+p)\sigma^2
=\boxed{\operatorname{MSPE}}.
$$
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