  A unitary family is a family of unitary operators $U(x)$ acting on a finite
dimensional hermitian vector space, depending analytically on a real parameter
$x$. It is monotone if $\frac1i U'(x)U(x)^{-1}$ is a positive operator for each
$x$. We prove a number of results generalizing standard theorems on the
spectral theory of a single unitary operator $U_0$, which correspond to the
'commutative' case $U(x)=e^{ix}U_0$. Also, for a two-parameter unitary family
-- for which there is no analytic perturbation theory -- we prove an implicit
function type theorem for the spectral data under the assumption that the
family is monotone in one argument.
