Unitary invariance and spectral variation

We call a norm on operators or matrices weakly unitarily invariant if its value at operator A is not changed by replacing A by U1AU, provided only that U is unitary. This class includes such norms as the numerical radius. We extend to all such norms an inequality that bounds the spectral variation when a normal operator A is replaced by another normal B in terms of the arclength of any normal path from A to B, computed using the norm in question. Related results treat the local metric geometry of the “manifold” of normal operators. We introduce a representation for weakly unitarily invariant matrix norms in terms of function norms over the unit ball, and identify this correspondence explicitly in certain cases.