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
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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 .
 [AAK]
V.
M. Adamjan, D.
Z. Arov, and M.
G. Kreĭn, Analytic properties of the Schmidt pairs of a
Hankel operator and the generalized SchurTakagi problem, Mat. Sb.
(N.S.) 86(128) (1971), 34–75 (Russian). MR 0298453
(45 #7505)
 [Ag1]
J. Agler, Interpolation, preprint (1987).
 [Ag2]
Jim
Agler, NevanlinnaPick interpolation on
Sobolev space, Proc. Amer. Math. Soc.
108 (1990), no. 2,
341–351. MR
986645 (90f:30041), http://dx.doi.org/10.1090/S00029939199009866452
 [AY]
J. Agler and N. J. Young, Functions which are almost multipliers of Hilbert function spaces, to appear in Proc. London Math. Soc. CMP 98:06
 [Ak]
N. I. Akhiezer, On a minimum problem in the theory of functions and on the number of roots of an algebraic equation which lie inside the unit circle, Izv. Akad. Nauk SSSR 9(1931) 11691189.
 [Ar]
N.
Aronszajn, Theory of reproducing
kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404. MR 0051437
(14,479c), http://dx.doi.org/10.1090/S00029947195000514377
 [BH]
Joseph
A. Ball and J.
William Helton, A BeurlingLax theorem for the Lie group
𝑈(𝑚,𝑛) which contains most classical interpolation
theory, J. Operator Theory 9 (1983), no. 1,
107–142. MR
695942 (84m:47046)
 [CG]
Tiberiu
Constantinescu and Aurelian
Gheondea, Minimal signature in lifting of operators. I, J.
Operator Theory 22 (1989), no. 2, 345–367. MR 1043732
(91b:47013)
 [CS]
Mischa
Cotlar and Cora
Sadosky, Nehari and NevanlinnaPick problems and holomorphic
extensions in the polydisk in terms of restricted BMO, J. Funct. Anal.
124 (1994), no. 1, 205–210. MR 1284610
(95f:47047), http://dx.doi.org/10.1006/jfan.1994.1105
 [DFT]
John
C. Doyle, Bruce
A. Francis, and Allen
R. Tannenbaum, Feedback control theory, Macmillan Publishing
Company, New York, 1992. MR 1200235
(93k:93002)
 [DGK]
Ph.
Delsarte, Y.
Genin, and Y.
Kamp, On the role of the NevanlinnaPick problem in circuit and
system theory, Internat. J. Circuit Theory Appl. 9
(1981), no. 2, 177–187. MR 612269
(82d:94052), http://dx.doi.org/10.1002/cta.4490090204
 [FF]
Ciprian
Foias and Arthur
E. Frazho, The commutant lifting approach to interpolation
problems, Operator Theory: Advances and Applications, vol. 44,
Birkhäuser Verlag, Basel, 1990. MR 1120546
(92k:47033)
 [G]
Keith
Glover, All optimal Hankelnorm approximations of linear
multivariable systems and their 𝐿^{∞}error bounds,
Internat. J. Control 39 (1984), no. 6,
1115–1193. MR 748558
(86a:93029), http://dx.doi.org/10.1080/00207178408933239
 [GRSW]
Israel
Gohberg, Leiba
Rodman, Tamir
Shalom, and Hugo
J. Woerdeman, Bounds for eigenvalues and singular values of matrix
completions, Linear and Multilinear Algebra 33
(1993), no. 34, 233–249. MR 1334675
(96h:15019), http://dx.doi.org/10.1080/03081089308818197
 [H]
J.
William Helton, Joseph
A. Ball, Charles
R. Johnson, and John
N. Palmer, Operator theory, analytic functions, matrices, and
electrical engineering, CBMS Regional Conference Series in
Mathematics, vol. 68, Published for the Conference Board of the
Mathematical Sciences, Washington, DC, 1987. MR 896034
(89f:47001)
 [K]
Irving
Kaplansky, Linear algebra and geometry. A second course, Allyn
and Bacon Inc., Boston, Mass., 1969. MR 0249444
(40 #2689)
 [MS]
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]
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
(94a:47026), http://dx.doi.org/10.1007/BF01358955
 [Q2]
P. Quiggin, Generalisations of Pick's Theorem to Reproducing Kernel Hilbert Spaces, Ph.D. thesis, Lancaster University, 1994.
 [S]
Donald
Sarason, Generalized interpolation in
𝐻^{∞}, Trans. Amer. Math.
Soc. 127 (1967),
179–203. MR 0208383
(34 #8193), http://dx.doi.org/10.1090/S00029947196702083838
 [AAK]
 V. M. Adamyan, D. Z. Arov and M. G. Krein, Analytic properties of Schmidt pairs of a Hankel operator and generalized SchurTakagi problem, Mat. Sbornik 86 (1971), 3373. MR 45:7505
 [Ag1]
 J. Agler, Interpolation, preprint (1987).
 [Ag2]
 J. Agler, NevanlinnaPick interpolation on Sobolev space, Proc. Amer. Math. Soc. 108(1990) 341351. MR 90f:30041
 [AY]
 J. Agler and N. J. Young, Functions which are almost multipliers of Hilbert function spaces, to appear in Proc. London Math. Soc. CMP 98:06
 [Ak]
 N. I. Akhiezer, On a minimum problem in the theory of functions and on the number of roots of an algebraic equation which lie inside the unit circle, Izv. Akad. Nauk SSSR 9(1931) 11691189.
 [Ar]
 N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337404. MR 14:479c
 [BH]
 J. A. Ball and J. W. Helton, A BeurlingLax theorem for the Lie group which contains most classical interpolation theory, J. Operator Theory 9(1983) 107142. MR 84m:47046
 [CG]
 T. Constantinescu and A. Gheondea, Minimal signature in lifting of operators I, J. Operator Theory 22 (1989) 345367. MR 91b:47013
 [CS]
 M. Cotlar and C. Sadosky, Nehari and NevalinnaPick problems and holomorphic extensions in the polydisk in terms of restricted BMO, J. Functional Analysis 124 (1994) 205210 . MR 95f:47047
 [DFT]
 J. C. Doyle, B. A. Francis and A. R. Tannenbaum, Feedback Control Theory, MaxwellMacmillan Publishing Co., New York 1992. MR 93k:93002
 [DGK]
 P. Delsarte, Y. Genin and Y. Kamp, On the role of the NevanlinnaPick problem in circuit and system theory, Circuit Theory and Applications 9 (1981) 177187. MR 82d:94052
 [FF]
 C. Foias and A. E. Frazho, The Commutant Lifting Approach to Interpolation Problems, OT44, Birkhäuser Verlag, Basel 1986. MR 92k:47033
 [G]
 K. Glover, All optimal Hankelnorm approximations of linear multivariable systems and their error bounds, Int. J. Control 39(1984) 11151193. MR 86a:93029
 [GRSW]
 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
 [H]
 J. W. Helton, Operator theory, analytic functions, matrices and electrical engineering,, CBMS Regional Conference Series No. 68, AMS, Providence 1987. MR 89f:47001
 [K]
 I. Kaplansky, Linear Algebra and Geometry, Chelsea Publishing Co., New York, 1969. MR 40:2689
 [MS]
 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
 [Q2]
 P. Quiggin, Generalisations of Pick's Theorem to Reproducing Kernel Hilbert Spaces, Ph.D. thesis, Lancaster University, 1994.
 [S]
 D. Sarason, Generalized interpolation in , Trans. Amer. Math. Soc. 127 (1967) 179203. MR 34:8193
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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
