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Summary: arXiv:math.OA/0606321v113Jun2006
DIAGONALS OF NORMAL OPERATORS
WITH FINITE SPECTRUM
WILLIAM ARVESON
Abstract. Let X = {1, . . . , N } 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 paper we take up the case in which X is the set of
vertices of a convex polygon in C. The critical sequences d turn out to
be those that accumulate rapidly in X in the sense that
n=1
dist (dn, X) < .
We show that there is an abelian group X a quotient of R2
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