A converse to a theorem of Adamyan, Arov and Krein
Authors:
J. Agler and N. J. Young
Journal:
J. Amer. Math. Soc. 12 (1999), 305333
MSC (1991):
Primary 46E22; Secondary 47B38
MathSciNet review:
1643649
Fulltext PDF Free Access
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Abstract: A well known theorem of Akhiezer, Adamyan, Arov and Krein gives a criterion (in terms of the signature of a certain Hermitian matrix) for interpolation by a meromorphic function in the unit disc with at most poles subject to an norm bound on the unit circle. One can view this theorem as an assertion about the Hardy space of analytic functions on the disc and its reproducing kernel. A similar assertion makes sense (though it is not usually true) for an arbitrary Hilbert space of functions. One can therefore ask for which spaces the assertion is true. We answer this question by showing that it holds precisely for a class of spaces closely related to .
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 C. Foias and A. E. Frazho, The Commutant Lifting Approach to Interpolation Problems, OT44, Birkhäuser Verlag, Basel 1986. MR 92k:47033
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 K. Glover, All optimal Hankelnorm approximations of linear multivariable systems and their error bounds, Int. J. Control 39(1984) 11151193. MR 86a:93029
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 I. Gohberg, L. Rodman, T. Shalom and H. J. Woerdemann, Bounds for eigenvalues and singular values of matrix completions, Linear and Multilinear Algebra 33 (1993) 233249. MR 96h:15019
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 D. E. Marshall and C. Sundberg, Interpolating sequences for multipliers of the Dirichlet space, to appear.
 [P]
 G. Pick, Über die Beschränkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden, Math. Ann. 77 (1916), 723.
 [Q1]
 P. Quiggin, For which reproducing kernel Hilbert spaces is Pick's theorem true? Integral Equations and Operator Theory 16 (1993), 244266. MR 94a:47026
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 P. Quiggin, Generalisations of Pick's Theorem to Reproducing Kernel Hilbert Spaces, Ph.D. thesis, Lancaster University, 1994.
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Additional Information
J. Agler
Affiliation:
Department of Mathematics, University of California at San Diego, La Jolla, California 92093
Email:
jagler@ucsd.edu
N. J. Young
Affiliation:
Department of Mathematics, University of Newcastle, Newcastle upon Tyne NE1 7RU, England
Email:
N.J.Young@ncl.ac.uk
DOI:
http://dx.doi.org/10.1090/S089403479900291X
PII:
S 08940347(99)00291X
Keywords:
Interpolation,
reproducing kernel,
multiplier,
Pick's theorem,
AdamyanArovKrein theorem,
Akhiezer's theorem
Received by editor(s):
May 28, 1997
Additional Notes:
J. Agler’s research was supported by an NSF grant in Modern Analysis.
Article copyright:
© Copyright 1999
American Mathematical Society
