Geodesic
A converse to a theorem of Adamyan, Arov and Krein
Agler, J.1 ; Young, N.2
1 Department of Mathematics, University of California at San Diego, La Jolla, California 92093
2 Department of Mathematics, University of Newcastle, Newcastle upon Tyne NE1 7RU, England
Journal of the American Mathematical Society, Tome 12 (1999) no. 2, pp. 305-333

Voir la notice de l'article provenant de la source American Mathematical Society

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 $m$ poles subject to an $L^\infty$-norm bound on the unit circle. One can view this theorem as an assertion about the Hardy space $H^2$ 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 $H^2$.
DOI : 10.1090/S0894-0347-99-00291-X

Agler, J. 1 ; Young, N. 2

1 Department of Mathematics, University of California at San Diego, La Jolla, California 92093
2 Department of Mathematics, University of Newcastle, Newcastle upon Tyne NE1 7RU, England 
@article{10_1090_S0894_0347_99_00291_X,
     author = {Agler, J. and Young, N.},
     title = {A converse to a theorem of {Adamyan,} {Arov} and {Krein}},
     journal = {Journal of the American Mathematical Society},
     pages = {305--333},
     publisher = {mathdoc},
     volume = {12},
     number = {2},
     year = {1999},
     doi = {10.1090/S0894-0347-99-00291-X},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-99-00291-X/}
}
TY  - JOUR
AU  - Agler, J.
AU  - Young, N.
TI  - A converse to a theorem of Adamyan, Arov and Krein
JO  - Journal of the American Mathematical Society
PY  - 1999
SP  - 305
EP  - 333
VL  - 12
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-99-00291-X/
DO  - 10.1090/S0894-0347-99-00291-X
ID  - 10_1090_S0894_0347_99_00291_X
ER  - 
%0 Journal Article
%A Agler, J.
%A Young, N.
%T A converse to a theorem of Adamyan, Arov and Krein
%J Journal of the American Mathematical Society
%D 1999
%P 305-333
%V 12
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-99-00291-X/
%R 10.1090/S0894-0347-99-00291-X
%F 10_1090_S0894_0347_99_00291_X
Agler, J.; Young, N. A converse to a theorem of Adamyan, Arov and Krein. Journal of the American Mathematical Society, Tome 12 (1999) no. 2, pp. 305-333. doi: 10.1090/S0894-0347-99-00291-X

[1] Adamjan, V. M., Arov, D. Z., Kreä­N, M. G. Analytic properties of the Schmidt pairs of a Hankel operator and the generalized Schur-Takagi problem Mat. Sb. (N.S.) 1971 34 75

[2] Agler, Jim Nevanlinna-Pick interpolation on Sobolev space Proc. Amer. Math. Soc. 1990 341 351

[3] Hebroni, P. Sur les inverses des éléments dérivables dans un anneau abstrait C. R. Acad. Sci. Paris 1939 285 287

[4] Ball, Joseph A., Helton, J. William A Beurling-Lax theorem for the Lie group 𝑈(𝑚,𝑛) which contains most classical interpolation theory J. Operator Theory 1983 107 142

[5] Constantinescu, Tiberiu, Gheondea, Aurelian Minimal signature in lifting of operators. I J. Operator Theory 1989 345 367

[6] Cotlar, Mischa, Sadosky, Cora Nehari and Nevanlinna-Pick problems and holomorphic extensions in the polydisk in terms of restricted BMO J. Funct. Anal. 1994 205 210

[7] Doyle, John C., Francis, Bruce A., Tannenbaum, Allen R. Feedback control theory 1992

[8] Delsarte, Ph., Genin, Y., Kamp, Y. On the role of the Nevanlinna-Pick problem in circuit and system theory Internat. J. Circuit Theory Appl. 1981 177 187

[9] Foias, Ciprian, Frazho, Arthur E. The commutant lifting approach to interpolation problems 1990

[10] Glover, Keith All optimal Hankel-norm approximations of linear multivariable systems and their 𝐿^{∞}-error bounds Internat. J. Control 1984 1115 1193

[11] Gohberg, Israel, Rodman, Leiba, Shalom, Tamir, Woerdeman, Hugo J. Bounds for eigenvalues and singular values of matrix completions Linear and Multilinear Algebra 1993 233 249

[12] Helton, J. William, Ball, Joseph A., Johnson, Charles R., Palmer, John N. Operator theory, analytic functions, matrices, and electrical engineering 1987

[13] Kaplansky, Irving Linear algebra and geometry. A second course 1969

[14] Quiggin, Peter For which reproducing kernel Hilbert spaces is Pick’s theorem true? Integral Equations Operator Theory 1993 244 266

[15] Sarason, Donald Generalized interpolation in 𝐻^{∞} Trans. Amer. Math. Soc. 1967 179 203

Cité par Sources :