A uniqueness result concerning Schur ideals
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- by James P. Solazzo PDF
- Proc. Amer. Math. Soc. 130 (2002), 1437-1445 Request permission
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
- J. Agler, Interpolation, preprint.
- William Arveson, Subalgebras of $C^{\ast }$-algebras. II, Acta Math. 128 (1972), no. 3-4, 271–308. MR 394232, DOI 10.1007/BF02392166
- Frank F. Bonsall and John Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973. MR 0423029, DOI 10.1007/978-3-642-65669-9
- Brian Cole, Keith Lewis, and John Wermer, Pick conditions on a uniform algebra and von Neumann inequalities, J. Funct. Anal. 107 (1992), no. 2, 235–254. MR 1172022, DOI 10.1016/0022-1236(92)90105-R
- Brian J. Cole and John Wermer, Pick interpolation, von Neumann inequalities, and hyperconvex sets, Complex potential theory (Montreal, PQ, 1993) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 439, Kluwer Acad. Publ., Dordrecht, 1994, pp. 89–129. MR 1332960
- Masamichi Hamana, Injective envelopes of operator systems, Publ. Res. Inst. Math. Sci. 15 (1979), no. 3, 773–785. MR 566081, DOI 10.2977/prims/1195187876
- R. Nevanlinna, Ueber beschrankte Funktionen, die in gegebene Punkten vorgeschriebene Werte annehmen, Ann. Acad. Sci. Fenn. (1919), 1-71.
- R. Nevanlinna, Ueber beschrankte analytische Funktionen, Ann. Acad. Sci. Fenn. #7 32 (1929).
- V.I. Paulsen, Matrix-valued interpolation and hyperconvex sets, Int. Eqn. and Op. Thy., to appear.
- V.I. Paulsen, Operator algebras of idempotents, preprint.
- G. Pick, Ueber die Beschraenkungen analytisher Funktionen, welche durch vorgegebene Funktionswete bewirkt werden, Math. Ann. 77 (1916), 7-23.
- Donald Sarason, Generalized interpolation in $H^{\infty }$, Trans. Amer. Math. Soc. 127 (1967), 179–203. MR 208383, DOI 10.1090/S0002-9947-1967-0208383-8
- Edgar Lee Stout, The theory of uniform algebras, Bogden & Quigley, Inc., Publishers, Tarrytown-on-Hudson, N.Y., 1971. MR 0423083
- A.T. Tomerlin, Products of Nevanlinna-Pick Kernels and Operator Colligations, Int. Eq. and Op. Thy. 38, 2000, 350–356.
Additional Information
- James P. Solazzo
- Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
- Email: solazzo@math.uga.edu
- 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
- © Copyright 2001 American Mathematical Society
- 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
- MathSciNet review: 1879967