  We give a short, operator-theoretic proof of the asymptotic independence
(including a first correction term) of the minimal and maximal eigenvalue of
the n \times n Gaussian Unitary Ensemble in the large matrix limit n \to
\infty. This is done by representing the joint probability distribution of the
extreme eigenvalues as the Fredholm determinant of an operator matrix that
asymptotically becomes diagonal. As a corollary we obtain that the correlation
of the extreme eigenvalues asymptotically behaves like n^{-2/3}/4\sigma^2,
where \sigma^2 denotes the variance of the Tracy--Widom distribution. While we
conjecture that the extreme eigenvalues are asymptotically independent for
Wigner random hermitian matrix ensembles in general, the actual constant in the
asymptotic behavior of the correlation turns out to be specific and can thus be
used to distinguish the Gaussian Unitary Ensemble statistically from other
Wigner ensembles.
