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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
DOI: https://doi.org/10.1090/S0002-9939-01-06211-6
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.


References [Enhancements On Off] (What's this?)

  • 1. J. Agler, Interpolation, preprint.
  • 2. W.B. Arveson, Subalgebras of $C^*$-Algebras II, Acta Math. 123 (1972), 271-308. MR 52:15035
  • 3. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer-Verlag, New York/Berlin, 1973. MR 54:11013
  • 4. B. Cole, K. Lewis and J. Wermer, Pick Conditions on a Uniform Algebra and von Neumann Inequalities, J. Functional Anal. 107 (1992), 235-254. MR 93e:46059
  • 5. B. Cole and J. Wermer, Pick Interpolation, von Neumann Inequalities, and Hyperconvex sets, Complex Potential Theory, P.M. Gauthier (ed.), NATO Adv. Sci. Inst. Ser. C 439, Kluwer, Dordrecht, (1994), 89-129. MR 96m:46092
  • 6. M. Hamana, Injective envelopes of Operator Systems, Publ. RIMS, Kyoto Univ. 15 (1979), 773-785. MR 81h:46071
  • 7. R. Nevanlinna, Ueber beschrankte Funktionen, die in gegebene Punkten vorgeschriebene Werte annehmen, Ann. Acad. Sci. Fenn. (1919), 1-71.
  • 8. R. Nevanlinna, Ueber beschrankte analytische Funktionen, Ann. Acad. Sci. Fenn. #7 32 (1929).
  • 9. V.I. Paulsen, Matrix-valued interpolation and hyperconvex sets, Int. Eqn. and Op. Thy., to appear.
  • 10. V.I. Paulsen, Operator algebras of idempotents, preprint.
  • 11. G. Pick, Ueber die Beschraenkungen analytisher Funktionen, welche durch vorgegebene Funktionswete bewirkt werden, Math. Ann. 77 (1916), 7-23.
  • 12. D. Sarason, Generalized interpolation in $H^\infty$, Trans. Amer. Math. Soc. 127 (1967), 179-203. MR 34:8193
  • 13. E.L. Stout, The Theory of Uniform Algebras, Bogden and Quigley, New York, 1971. MR 54:11066
  • 14. A.T. Tomerlin, Products of Nevanlinna-Pick Kernels and Operator Colligations, Int. Eq. and Op. Thy. 38, 2000, 350-356. CMP 2001:05

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Additional Information

James P. Solazzo
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: solazzo@math.uga.edu

DOI: https://doi.org/10.1090/S0002-9939-01-06211-6
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

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