  It is shown that certain ensembles of random matrices with entries that
vanish outside a band around the diagonal satisfy a localization condition on
the resolvent which guarantees that eigenvectors have strong overlap with a
vanishing fraction of standard basis vectors, provided the band width $W$
raised to a power $\mu$ remains smaller than the matrix size $N$. For a
Gaussian band ensemble, with matrix elements given by i.i.d. centered Gaussians
within a band of width $W$, the estimate $\mu \le 8$ holds.
