Abstract
The ultimate goal of our campaign is to generalize a substantial collection of results in classical complex variables to highly nonlinear situations. In [BH1] and subsequent works (c. f. [BGR],[H]) it was shown how an extension of the classical Beurling-Lax-Halmos theorem to Hilbert spasces with a signed bilinear form (indefinite metric) could be regarded as the key to many theorems in complex analysis. Thus our approach to the nonlinear case is to first extend our indefinite metric Beurling-Lax-Halmos theory to nonlinear situations that is the subject of this article.
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References
[AD] V. Anantharam and C. A. Desoer, On the stabilization of nonlinear systems, IEEE Trans. Auto. Control,AC-29 (1984), 569–573.
[Be] M. Berger,Nonlinearity and Functional Analysis: Lectures on Nonlinear Problems in Functional Analysis, Academic Press, New York, 1977.
[BFHT1] J. A. Ball, C. Foias, J. W. Helton and A. Tannenbaum, A Poincaré-Dulac approach to a nonlinear Beurling-Lax-Halmos theorem, Math. Anal. Appl., to appear.
[BFHT2] J. A. Ball, C. Foias, J. W. Helton and A. Tannenbaum, On a local nonlinear commutant lifting theorem, Indiana Univ. Math. J.,36 (1987), 693–709.
[BH1] J. A. Ball and J. W. Helton, A Beurling-Lax theorem for the Lie groupU(m, n) which contains most classical interpolation theory, J. Operator Theory,9 (1983), 107–142.
[BH2] J. A. Ball and J. W. Helton, in preparation.
[BH3] J. A. Ball and J. W. Helton, Interpolation problems of Pick-Nevanlinna and Löewner types for meromorphic matrix functions: parametrization of the set of all solutions, Integral Equations and Operator Theory,9 (1986), 156–203.
[BH4] J. A. Ball and J. W. Helton, Shift invariant subspaces, passivity, reproducing kernels andH ∞-optimization, in Proceedings of Conference on Operator Theory and Applications (Mesa, Arizona, 1987), Birkhäuser, to appear.
[BH5] J. A. Ball and J. W. Helton, Sensitivity bandwidth optimization for nonlinear feedback systems, Proceedings of Math. Theory of Networks and Systems Conference (Phoenix, 1987), to appear.
[BHS] J. A. Ball, J. W. Helton and C. H. Sung, Nonlinear solutions of Nevanlinna-Pick interpolation problems, Michigan Math. J.,34 (1987), 375–389.
[BGR] J. A. Ball, I. Gohberg and L. Rodman,Interpolation Problems for Matrix Valued Functions and Applications, Part I: Rational Functions, monograph, in preparation.
[CL] E. Coddington and N. Levinson,Theory of Linear Differential Equations McGraw-Hill, New York.
[DL] C. A. Desoer and R.-W. Liu, Global parametrazation of feedback systems with nonlinear plants, Systems and Control Letters,1, (1982), 249–251.
[Du] J. Dugundji,Topology, Allyn and Bacon, Boston, 1966.
[F] B. A. Francis,A Course in H ∞ Control Theory, Lecture Notes in Control and Inf. Sci., Springer-Verlag, New York-Heidelberg-Berlin, 1987.
[GL] I. Gohberg and J. Leiterer, Cocycles, operator-valued functions and families of subspaces, Math. Issled. (Kishinev)8 (1973), 23–55. [In Russian.]
[GS] V. Guillemin and S. Sternberg, Geometric Asymptotics, Math. Surveys, No. 14, Amer. Math. Soc., Providence, RI, 1977.
[H] J. W. Helton, Operator theory, analytic functions, matrices and electrical engineering, CBMS #68, Amer. Math. Soc., Providence, 1987.
[K] T. Kailath,Linear Systems, Prentice-Hall, Englewood Cliffs, 1980.
[L] S. Lange,Introduction to Differentiable Manifolds, Interscience Publishers, New York.
[M] J. Milnor,Morse Theory, Princeton Univ. Press, Princeton, 1963.
[S] S. Smale, Morse theory and a nonlinear generalization of the Dirichlet problem, Annals of Math.,80 (1964), 382–396.
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Supported in part by the Air Force Office of Scientific Research, the National Science Foundation and the Office of Naval Research.
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Ball, J.A., Helton, J.W. Shift invariant manifolds and nonlinear analytic function theory. Integr equ oper theory 11, 615–725 (1988). https://doi.org/10.1007/BF01207393
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DOI: https://doi.org/10.1007/BF01207393