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Diagonals of Normal Operators with Finite Spectrum

William Arveson
Proceedings of the National Academy of Sciences of the United States of America
Vol. 104, No. 4 (Jan. 23, 2007), pp. 1152-1158
Stable URL: http://www.jstor.org/stable/25426253
Page Count: 7
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Diagonals of Normal Operators with Finite Spectrum
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Abstract

Let $X=\{\lambda _{1},\ldots,\lambda _{N}\}$ 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 $d_{n}$ 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 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 $\sum_{n=1}^{\infty}\text{dist}(d_{n},X)<\infty $. We show that there is an abelian group Γₓ, a quotient of ${\Bbb R}^{2}$ 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) ≠ 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.

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