Diagonals of normal operators with finite spectrum.

Abstract

Let X={lambda1, ..., lambdaN} be a finite set of complex numbers, and let A be a normal operator with spectrum X that acts on a separable Hilbert space H. Relative to a fixed orthonormal basis e1, e2, ... for H, A gives rise to a matrix whose diagonal is a sequence d=(d1, d2, ...) with the property that each of its terms dn belongs to the convex hull of X. Not all sequences with that property can arise as the diagonal of a normal operator with spectrum X. The case where X is a set of real numbers has received a great deal of attention over the years and is reasonably well (though incompletely) understood. In this work we take up the case in which X is the set of vertices of a convex polygon in . The critical sequences d turn out to be those that accumulate rapidly in X in the sense that Sigmainfinityn=1 dist(dn, X)<infinity. We show that there is an abelian group GammaX, a quotient of R2 by a countable subgroup with concrete arithmetic properties, and a subjective mapping of such sequences d-->s(d)E GammaX with the following property: If s(d) not equal 0, then d is not the diagonal of any such operator A. We also show that while this is the only obstruction when N=2, there are other (as yet unknown) obstructions when N=3.