Discrete-time dichotomous well-posed linear systems and generalized Schur-Nevanlinna-Pick interpolation

Complex Analysis and Operator Theory

Volumn 1, Issue 1, 2007, Pages 1-54

  Ball, Joseph A ^a   Raney, Michael W ^a  

^a VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY   (United States)

Author keywords

Dichotomous realization; Grassmannian approach to interpolation; Shift invariant subspace; Strongly regular J inner function

Indexed keywords

EID: 34250851611     PISSN: 16618254     EISSN: 16618262     Source Type: Journal    
DOI: 10.1007/s11785-006-0001-y     Document Type: Article

References (42)

1. D. Alpay and I. Gohberg, Unitary rational matrix functions, in Topics in Interpolation Theory and Rational Matrix-Valued Functions (Ed. I. Gohberg), pp. 175-222, OT 33, Birkhäuser-Verlag, Basel-Boston, 1988.
2. D. Z. Arov, γ-generating matrices, j-inner matrix-functions and related extrapolation problems, Teor. Funktsii Funktsional. Anal. i Prilozhen., I, 51 (1989), 61-67; II, 52 (1989), 103-109; III, 53 (1990), 57-64; English transl., J. Soviet Math. I, 52 (1990), 3487-3491; II, 52 (1990), 342-3425; III, 58 (1992), 532, 537.
3. D. Z. Arov, γ-generating matrices, j-inner matrix-functions and related extrapolation problems, IV, Mathematical Physics, Analysis, Geometry 2 (1995), 3-14.
4. D. Z. Arov and H. Dym, J-inner matrix functions, interpolation and inverse problems for canonical systems, I: Foundations, Integral Equations and Operator Theory 29 (1997), 373-454.
5. D. Z. Arov and H. Dym, J-inner matrix functions, interpolation and inverse problems, II: The inverse monodromy problem, Integral Equations and Operator Theory 36 (2000), 11-70.
6. D. Z. Arov and H. Dym, J-inner matrix functions, interpolation and inverse problems for canonical systems, III: More on the inverse monodromy problem, Integral Equations and Operator Theory 36 (2000), 127-181.
7. D. Z. Arov and H. Dym, J-inner matrix functions, interpolation and inverse problems for canonical systems, IV: Direct and inverse bitangential input scattering problem, Integral Equations and Operator Theory 43 (2002), 1-67.
8. D. Z. Arov and H. Dym, J-inner matrix functions, interpolation and inverse problems for canonical systems, V: The inverse input scattering problem for Wiener class and rational p × q input scattering matrices, Integral Equations and Operator Theory 43 2002, 68-129.
9. D. Z. Arov and H. Dym, Matricial Nehari problems, J-inner matrix functions and the Muckenhoupt condition, J. Functional Analysis 280 (2003), 387-399.
10. D. Z. Arov and H. Dym, Matricial Nehari problems, J-inner matrix functions and γ-generating matrices, J. Math. Anal. Appl. 280 (2003), 387-399.
11. D. Z. Arov and H. Dym, Strongly regular J-inner matrix functions and related problems, in Current Trends in Operator Theory and its Applications (Ed. J. A. Ball, J.W. Helton, J. Klaus and L. Rodman), pp. 79-106, OT 149, Birkhäuser-Verlag, Basel-Boston, 2004.
12. D. Z. Arov and H. Dym, Strongly regular J-inner matrix valued functions and inverse problems for canonical systems, Recent Advances in Operator Theory and its Applications (Ed. M. A. Kaashoek, S. Seat zu and C. van der Mee), pp. 101-160, OT 160, Birkhäuser-Verlag, Basel-Boston, 2005.
13. D. Z. Arov and O. J. Staffans, State/signal linear time-invariant systems theory, Part I: Discrete time systems, in The State Space Method Generalizations and Applications (Ed. D. Alpay and I. Gohberg), pp. 115-177, OT 161, Birkhäuser-Verlag, Basel-Boston-Berlin, 2005.
14. Z. D. Arova, J-unitary nodes with strongly regular J-inner characteristic matrix functions, Methods in Functional Analysis and Topology 6 (2000), no. 3, 9-23.
15. Z. D. Arova, On Livšic-Brodskii nodes with strongly regular J-inner characteristic matrix functions, in Recent Advances in Operator Theory and Related Topics (Ed. L. Kérchy, C. Foias, I. Gohberg and H. Langer) pp. 83-96, OT 127, Birkhäuser-Verlag, Basel-Boston, 2001.
16. Z. D. Arova, Operator Nodes with Strongly Regular Characteristic Function, Academisch Proefschrift, Vrije Universiteit, Amsterdam, 2003.
17. J. A. Ball, Nevanlinna-Pick interpolation: generalizations and applications, in Recent Results in Operator Theory, Vol. I (Ed. J.B. Conway and B.B. Morrel), pp. 51-94, Longman Sci. Tech., Harlow, 1988.
18. J. A. Ball, N. Cohen and A. C. M. Ran, Inverse spectral problems for regular improper rational matrix functions, in Topics in Interpolation Theory and Rational Matrix-Valued Functions (Ed. I. Gohberg), pp. 123-173, OT 33, Birkhäuser-Verlag, Basel-Boston, 1988.
19. J. A. Ball, I. Gohberg and L. Rodman, Interpolation of Rational Matrix Functions, OT 45, Birkhäuser-Verlag, Basel-Boston, 1990.
20. J.A. Ball and J. W. Helton, A Beurling-Lax theorem for the Lie group U(m,n) which contains most classical interpolation, J. Operator Theory 9 (1983), 107-142.
21. J. A. Ball and J. W. Helton, Beurling-Lax representations using classical Lie groups with many applications II: GL(n, ℂ) and Wiener-Hopf factorization, Integral Equations and Operator Theory 7 (1984), no. 3, 291-309.
22. ∞-optimization, in Contributions to Operator Theory and its Applications (Ed. J.W. Helton and L. Rodman), pp. 265-310, OT 35, Birkhäuser- Verlag, Basel-Boston, 1988.
23. J. A. Ball, K. M. Mikkola and A. J. Sasane, State-space formulas for the Nehari-Takagi problem for nonexponentially stable infinite-dimensional systems, SIAM J. Control Optim. 44 (2006), no. 2, 531-563.
24. J. A. Ball and A. C. M. Ran, Global inverse spectral problems for rational matrix functions, Linear Algebra and its Applications 86 (1987), 237-282.
25. J. A. Ball and A. C. M. Ran, Local inverse spectral problems for rational matrix functions, Integral Equations and Operator Theory 10 (1987), 349-415.
26. A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1949), 239-255.
27. J. Bognár, Indefinite Inner Product Spaces, Springer-Verlag, New York-Heidelberg-Berlin, 1974.
28. R. G. Douglas and J. W. Helton, Inner dilations of analytic matrix functions and Darlington synthesis, Acta Sci. Math. (Szeged) 34 (1973), 61-67.
29. R. G. Douglas, H.S. Shapiro and A.L. Shields, Cyclic vectors of the backward shift, Ann. Inst. Fourier (Grenoble) 20 (1970), 37-76.
30. H. Dym, J-Contractive Matrix Functions, Reproducing Kernel Hilbert Spaces and Interpolation, CBMS Regional Conference Series 71, American Mathematical Society, Providence, 1989.
31. C. Foias and A. E. Frazho, The Commutant Lifting Approach to Interpolation Problems, OT 44, Birkhäuser-Verlag, Basel-Boston, 1990.
32. C. Foias, A. E. Frazho, I. Gohberg and M. A. Kaashoek, Metric Constrained Interpolation, Commutant Lifting and Systems, OT 100, Birkhäuser-Verlag, Basel-Boston, 1998.
33. P. R. Halmos, Shifts on Hilbert spaces, J. Reine Angew. Math. 208 (1961), 102-112.
34. J. W. Helton, J. A. Ball, C.B. Johnson and J.N. Palmer, Operator Theory, Analytic Functions, Matrices and Electrical Engineering, CBMS Regional Conference Series 68, American Mathematical Society, Providence, 1987.
35. O. Iftume, M. A. Kaashoek and A. Sasane, A Grassmannian band method approach to the Nehari-Takagi problem, J. Math. Anal. Appl. 310 (2005), 97-115.
36. P. D. Lax, Translation invariant spaces, Acta Math. 101 (1959), 163-178.
37. J. Malinen, O. J. Staffans and G. Weiss, When is a linear system conservative? Quarterly of Applied Mathematics LXIV (2006), no. 1, 61-91.
38. B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North Holland Publishing Company, Amsterdam-London, 1970.
39. A. A. Nudelman, On a new problem of moment type, Soviet Math. Doklady 18 (1977), 507-510.
40. M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, Oxford University Press, New York, 1985.
41. W. Rudin, Functional Analysis, Second Edition, International Series in Pure and Applied Mathematics, McGraw Hill, Boston, 1991.
42. O. J. Staffans, Well-Posed Linear Systems, Cambridge University Press, Cambridge, 2004.