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Southeastern Analysis Meeting
March 5-9, 2008
Vanderbilt University
Nashville, Tennessee, USA

Organizers
Brett Wick, Daoxing Xia, Dechao Zheng

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Matrices and Varieties
by
John McCarthy
Washington University
Coauthors: Jim Agler

For any pair T = (T1, T2) of commuting matrices, normalized so that both have norm one, there are many polynomials p(z1, z2) that annihilate the pair. There is a special choice with the property that the set V = { (z1, z2) : |z1| ≤ 1, |z2| ≤ 1,  p(z1, z2) = 0 } is a spectral set for T, i.e. for any other polynomial q the inequality
∥ q(T1, T2) ∥ ≤ ∥ q ∥V
holds.

I shall discuss how these bordered varieties V arise, and, more generally, some connections between the geometry of varieties and properties of function algebras.

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Date received: January 14, 2008

Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cavq-11.