A simple proof of necessity in the McCullough-Quiggin theorem
HTML articles powered by AMS MathViewer
- by Greg Knese PDF
- Proc. Amer. Math. Soc. 148 (2020), 3453-3456 Request permission
Abstract:
A short and simple proof of necessity in the McCullough-Quiggin characterization of positive semi-definite kernels with the complete Pick property is presented.References
- Jim Agler and John E. McCarthy, Complete Nevanlinna-Pick kernels, J. Funct. Anal. 175 (2000), no. 1, 111–124. MR 1774853, DOI 10.1006/jfan.2000.3599
- Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, vol. 44, American Mathematical Society, Providence, RI, 2002. MR 1882259, DOI 10.1090/gsm/044
- Alexandru Aleman, Michael Hartz, John E. McCarthy, and Stefan Richter, Interpolating sequences in spaces with the complete Pick property, Int. Math. Res. Not. IMRN 12 (2019), 3832–3854. MR 3973111, DOI 10.1093/imrn/rnx237
- Joseph A. Ball, Tavan T. Trent, and Victor Vinnikov, Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces, Operator theory and analysis (Amsterdam, 1997) Oper. Theory Adv. Appl., vol. 122, Birkhäuser, Basel, 2001, pp. 89–138. MR 1846055
- Raphaël Clouâtre and Michael Hartz, Multiplier algebras of complete Nevanlinna-Pick spaces: dilations, boundary representations and hyperrigidity, J. Funct. Anal. 274 (2018), no. 6, 1690–1738. MR 3758546, DOI 10.1016/j.jfa.2017.10.008
- Peter Quiggin, For which reproducing kernel Hilbert spaces is Pick’s theorem true?, Integral Equations Operator Theory 16 (1993), no. 2, 244–266. MR 1205001, DOI 10.1007/BF01358955
- Michael T. Jury and Robert T. W. Martin, Factorization in weak products of complete pick spaces, Bull. Lond. Math. Soc. 51 (2019), no. 2, 223–229. MR 3937583, DOI 10.1112/blms.12222
- Gregory Marx, The complete Pick property and reproducing kernel Hilbert spaces, Master’s Thesis, Virginia Tech, 2013.
- John E. McCarthy, Pick’s theorem—what’s the big deal?, Amer. Math. Monthly 110 (2003), no. 1, 36–45. MR 1952746, DOI 10.2307/3072342
- Scott McCullough, Carathéodory interpolation kernels, Integral Equations Operator Theory 15 (1992), no. 1, 43–71. MR 1134687, DOI 10.1007/BF01193766
- Scott McCullough, The local de Branges-Rovnyak construction and complete Nevanlinna-Pick kernels, Algebraic methods in operator theory, Birkhäuser Boston, Boston, MA, 1994, pp. 15–24. MR 1284929
- Vern I. Paulsen and Mrinal Raghupathi, An introduction to the theory of reproducing kernel Hilbert spaces, Cambridge Studies in Advanced Mathematics, vol. 152, Cambridge University Press, Cambridge, 2016. MR 3526117, DOI 10.1017/CBO9781316219232
- Georg Pick, Über die Beschränkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden, Math. Ann. 77 (1915), no. 1, 7–23 (German). MR 1511844, DOI 10.1007/BF01456817
Additional Information
- Greg Knese
- Affiliation: Department of Mathematics and Statistics, Washington University in St. Louis, One Brookings Drive, St. Louis, Missouri 63130
- MR Author ID: 813491
- Email: geknese@wustl.edu
- Received by editor(s): December 30, 2019
- Published electronically: April 9, 2020
- Additional Notes: The author was partially supported by NSF grant DMS-1900816
- Communicated by: Stephan Ramon Garcia
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 3453-3456
- MSC (2010): Primary 46E22, 47A57
- DOI: https://doi.org/10.1090/proc/15061
- MathSciNet review: 4108851