Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The two-by-two spectral Nevanlinna-Pick problem


Authors: Jim Agler and N. J. Young
Journal: Trans. Amer. Math. Soc. 356 (2004), 573-585
MSC (2000): Primary 30E05
DOI: https://doi.org/10.1090/S0002-9947-03-03083-6
Published electronically: September 22, 2003
MathSciNet review: 2022711
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give a criterion for the existence of an analytic $2 \times 2$matrix-valued function on the disc satisfying a finite set of interpolation conditions and having spectral radius bounded by $1$. We also give a realization theorem for analytic functions from the disc to the symmetrised bidisc.


References [Enhancements On Off] (What's this?)

  • 1. J. Agler, F. B. Yeh and N. J. Young, The symmetrised bidisc, complex geodesics and realizations, to appear in Reproducing Kernel Hilbert Spaces, Positivity, Function Theory, System Theory and Related Topics, D. Alpay, ed., Birkhäuser OT series.
  • 2. J. Agler and N. J. Young, A Schwarz lemma for the symmetrized bidisc, Bull. London Math. Soc., 33 (2001) 175-186. MR 2002e:30026
  • 3. J. Agler and N. J. Young, The two-point spectral Nevanlinna-Pick problem, Integral Equations Operator Theory 37 (2000) 375-385. MR 2001g:47025
  • 4. J. A. Ball, I. C. Gohberg and L. Rodman, Interpolation of Rational Matrix Functions, Operator Theory: Advances and Applications, Vol. 45, Birkhäuser, Basel, 1990. MR 92m:47027
  • 5. J. A. Ball and N. J. Young, Problems on the realization of functions, Fields Inst. Communications 25 (2000) 179-185 (the Proceedings of a conference on applied operator theory, Winnipeg, 1998). MR 2001a:47001
  • 6. H. Bercovici, C. Foias and A. Tannenbaum, A spectral commutant lifting theorem, Trans. Amer. Math. Soc. 325 (1991) 741-763. MR 91j:47006
  • 7. H. Bercovici, C. Foias and A. Tannenbaum, Spectral variants of the Nevanlinna-Pick interpolation problem, Signal processing, scattering and operator theory, and numerical methods (Amsterdam, 1989), 23-45, Progr. Systems Control Theory, 5, Birkhäuser Boston, Boston, MA, 1990. MR 92k:47031
  • 8. H. Bercovici, C. Foias and A. Tannenbaum, The structured singular value for linear input/output operators, SIAM J. Control Optim, 34 (1996) 1392-1404. MR 97g:93030
  • 9. H. Bercovici, C. Foias and A. Tannenbaum, On the structured singular value for operators on Hilbert space. Feedback control, nonlinear systems, and complexity (Montreal, PQ, 1994), 11-23, Lecture Notes in Control and Inform. Sci., 202, Springer, London, 1995. MR 96b:47068
  • 10. J. C. Doyle, Analysis of feedback systems with structured uncertainties. Proc. IEE-D 129 (1982), no. 6, 242-250. MR 84a:93035
  • 11. J. C. Doyle and A. Packard, The complex structured singular value, Automatica J. IFAC, 29 (1993) 71-109. MR 93k:93042
  • 12. P. Duren, Theory of $H^p$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 42:3552
  • 13. A. Feintuch and A. Markus, The structured norm of a Hilbert space operator with respect to a given algebra of operators. Operator theory and interpolation (Bloomington, IN, 1996), 163-183, Oper. Theory Adv. Appl., Vol. 115, Birkhäuser, Basel, 2000. MR 2001h:47013
  • 14. Matlab $\mu$-Analysis and Synthesis Toolbox, The Math Works Inc., Natick, Massachusetts (http://www.mathworks.com/products/muanalysis/).
  • 15. A. Nokrane and T. J. Ransford, Schwarz's Lemma for algebroid multifunctions, Complex Variables: Theory and Application 45 (2001) 183-196. MR 2003f:30028
  • 16. S. Petrovic, An extremal problem in interpolation theory. Houston J. Math. 26 (2000) 165-181. MR 2001k:30050
  • 17. I. Schur, On power series which are bounded in the interior of the unit circle I, II, (English translation) in Operator Theory: Advances and Applications Vol. 18, I. Schur methods in operator theory and signal processing, 31-59 and 61-88, Birkhäuser, Basel 1986. MR 89c:00052
  • 18. B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, Akadémiai Kiadó, Budapest, 1970. MR 43:947
  • 19. J. L. Walsh, Interpolation and approximation by rational functions in the complex domain. Fourth edition. American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I. 1965. MR 36:1672b

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 30E05

Retrieve articles in all journals with MSC (2000): 30E05


Additional Information

Jim Agler
Affiliation: Department of Mathematics, University of California at San Diego, La Jolla, California 92093

N. J. Young
Affiliation: School of Mathematics and Statistics, University of Newcastle upon Tyne, Merz Court, Newcastle upon Tyne NE1 7RU, England

DOI: https://doi.org/10.1090/S0002-9947-03-03083-6
Received by editor(s): October 9, 2001
Received by editor(s) in revised form: February 26, 2002
Published electronically: September 22, 2003
Additional Notes: This research was supported by an NSF grant in Modern Analysis and an EPSRC Visiting Fellowship
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society