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A uniqueness result concerning Schur ideals

Author: James P. Solazzo
Journal: Proc. Amer. Math. Soc. 130 (2002), 1437-1445
MSC (2000): Primary 47A57, 46L07; Secondary 47L25, 47L30
Published electronically: October 12, 2001
MathSciNet review: 1879967
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Abstract: A subset of the set of all positive semi-definite matrices of a given size which is invariant under Schur (componentwise) multiplication by an arbitrary positive semi-definite matrix is said to be a Schur ideal. A subset of $k$-dimensional complex space is said to be $hyperconvex$ if it arises as the set of possible values $(w_{1}, \dots, w_{k}) = (f(\alpha_{1}), \dots, f(\alpha_{k}))$arising from restricting contractive elements $f$ from some uniform algebra $A$ to a finite set $\{ \alpha_{1}, \dots, \alpha_{k} \}$ in the domain. When the uniform algebra is the disk algebra, the hyperconvex set is said to be a Pick body. Motivated by the classical Pick interpolation theorem, Paulsen has introduced a natural notion of duality between Schur ideals and hyperconvex sets. By using some recently developed results in operator algebras (matricial Schur ideals), we show that each Pick body has a unique affiliated Schur ideal.

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Additional Information

James P. Solazzo
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602

Received by editor(s): October 4, 2000
Received by editor(s) in revised form: November 20, 2000
Published electronically: October 12, 2001
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2001 American Mathematical Society