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 $\frac{1}{i}{U}^{\prime }(x)U(x{)}^{-1}$ 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.

9 pages; extended version of what was the appendix to arXiv:0710.3405 v1