Monotone unitary families

Abstract:

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 $\frac 1i U’(x)U(x)^{-1}$ 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 $U(x)=e^{ix}U_0$. So these may be viewed as a noncommutative generalization of the spectral theory of $U_0$. 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.