## A converse to a theorem of Adamyan, Arov and Krein

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- by J. Agler and N. J. Young PDF
- J. Amer. Math. Soc.
**12**(1999), 305-333 Request permission

## 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 $m$ 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 $H^2$.

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## Additional Information

**J. Agler**- Affiliation: Department of Mathematics, University of California at San Diego, La Jolla, California 92093
- MR Author ID: 216240
- 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
- Received by editor(s): May 28, 1997
- Additional Notes: J. Agler’s research was supported by an NSF grant in Modern Analysis.
- © Copyright 1999 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**12**(1999), 305-333 - MSC (1991): Primary 46E22; Secondary 47B38
- DOI: https://doi.org/10.1090/S0894-0347-99-00291-X
- MathSciNet review: 1643649