Monotone unitary families

Daniel Grieser
2012 Proceedings of the American Mathematical Society  
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 1 i U ′ (x)U(x) −1 is a positive operator for each x. We pr ove a numb er of r es ul ts gener al i zi ng s tandar d theor em s on the s p ectr al 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
more » ... 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. 2000 Mathematics Subject Classification. Primary 47A55 .
doi:10.1090/s0002-9939-2012-11552-7 fatcat:akbj6tltljfv3fp25n2an64upq