The two-by-two spectral Nevanlinna-Pick problem

Agler, Jim; Young, N.

doi:10.1090/S0002-9947-03-03083-6

The two-by-two spectral Nevanlinna-Pick problem
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by Jim Agler and N. J. Young PDF

Trans. Amer. Math. Soc. 356 (2004), 573-585 Request permission

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

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.
J. Agler and N. J. Young, A Schwarz lemma for the symmetrized bidisc, Bull. London Math. Soc. 33 (2001), no. 2, 175–186. MR 1815421, DOI 10.1112/blms/33.2.175
J. Agler and N. J. Young, The two-point spectral Nevanlinna-Pick problem, Integral Equations Operator Theory 37 (2000), no. 4, 375–385. MR 1780117, DOI 10.1007/BF01192826
Joseph A. Ball, Israel Gohberg, and Leiba Rodman, Interpolation of rational matrix functions, Operator Theory: Advances and Applications, vol. 45, Birkhäuser Verlag, Basel, 1990. MR 1083145, DOI 10.1007/978-3-0348-7709-1
J. A. Ball and N. J. Young, Problems on the realization of functions, Operator theory and its applications (Winnipeg, MB, 1998) Fields Inst. Commun., vol. 25, Amer. Math. Soc., Providence, RI, 2000, pp. 179–185. MR 1759542
Hari Bercovici, Ciprian Foias, and Allen Tannenbaum, A spectral commutant lifting theorem, Trans. Amer. Math. Soc. 325 (1991), no. 2, 741–763. MR 1000144, DOI 10.1090/S0002-9947-1991-1000144-9
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) Progr. Systems Control Theory, vol. 5, Birkhäuser Boston, Boston, MA, 1990, pp. 23–45. MR 1115441
Hari Bercovici, Ciprian Foias, and Allen Tannenbaum, The structured singular value for linear input/output operators, SIAM J. Control Optim. 34 (1996), no. 4, 1392–1404. MR 1395840, DOI 10.1137/S0363012994268825
Hari Bercovici, Ciprian Foias, and Allen Tannenbaum, On the structured singular value for operators on Hilbert space, Feedback control, nonlinear systems, and complexity (Montreal, PQ, 1994) Lect. Notes Control Inf. Sci., vol. 202, Springer, London, 1995, pp. 11–23. MR 1326150, DOI 10.1007/BFb0027667
John Doyle, Analysis of feedback systems with structured uncertainties, Proc. IEE-D 129 (1982), no. 6, 242–250. MR 685109, DOI 10.1049/ip-d.1982.0053
A. Packard and J. Doyle, The complex structured singular value, Automatica J. IFAC 29 (1993), no. 1, 71–109. MR 1200542, DOI 10.1016/0005-1098(93)90175-S
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
Avraham Feintuch and Alexander Markus, The structured norm of a Hilbert space operator with respect to a given algebra of operators, Operator theory and interpolation (Bloomington, IN, 1996) Oper. Theory Adv. Appl., vol. 115, Birkhäuser, Basel, 2000, pp. 163–183. MR 1766812
Matlab $\mu$-Analysis and Synthesis Toolbox, The Math Works Inc., Natick, Massachusetts (http://www.mathworks.com/products/muanalysis/).
Abdelkrim Nokrane and Thomas Ransford, Schwarz’s lemma for algebroid multifunctions, Complex Variables Theory Appl. 45 (2001), no. 2, 183–196. MR 1909433, DOI 10.1080/17476930108815376
Srdjan Petrovic, An extremal problem in interpolation theory, Houston J. Math. 26 (2000), no. 1, 165–181. MR 1814733
I. Schur, On power series which are bounded in the interior of the unit circle. I, II, I. Schur methods in operator theory and signal processing, Oper. Theory Adv. Appl., vol. 18, Birkhäuser, Basel, 1986, pp. 31–59, 61–88. Translated from the German. MR 902602, DOI 10.1007/978-3-0348-5483-2_{3}
Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190
J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I., 1960. MR 0218587