Diagonals of normal operators with finite spectrum

Abstract

Let X 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 e_1,e_2, ... for H, A gives rise to a matrix whose diagonal is a sequence d=(d_1,d_2,...) with the property that each of its terms d_n 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 paper we take up the case in which X is the set of vertices of a convex polygon in the complex plane. The critical sequences d turn out to be those that accumulate rapidly in X in the sense that $$ \sum_{n=1}^\infty {\rm{dist}} (d_n,X)<\infty. $$ We show that there is an abelian group $\Gamma_X$ -- a quotient of $R^2$ by a countable subgroup with concrete arithmetic properties -- and a surjective mapping of such sequences $d\mapsto s(d)\in\Gamma_X$ with the following property: If s(d) is not 0, then d is not the diagonal of any such operator A. We also show that while this is the only obstruction when X contains two points, there are other (as yet unknown) obstructions when X contains more than two points.