Abstract
The state space theory from linear control theory is used as a tool to describe the action of automorphisms of a certain form acting on rational matrix functions. This class consists of automorphisms representable as the composition of a linear fractional change of variable together with the operations of inverse-transpose, conjugation and an inner automorphism. We also describe in state space terms minimal factorizations within the class of functions of a certain associated group.
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[ABGR] D. Alpay, J.A. Ball, I. Gohberg and L. Rodman, Realizations and factorizations of rational matrix functions with symmetries,Operator Theory: Advances and Applications 47 (1990), 1–60.
[BGR1] J.A. Ball, I. Gohberg and L. Rodman,Interpolation of Rational Matrix Functions, OT 45, Birkhäuser-Verlag, Boston, Basel, 1990.
[BGR2] J.A. Ball, I. Gohberg and L. Rodman, Sensitivity minimization and bitangential Nevanlinna-Pick interpolation in the contour integral form,IMA, Volume 23 (F.A. Grunbaum, J.W. Helton and P. Khargonekar, eds.), 1990, pp. 3–35.
[BGR3] J.A. Ball, I. Gohberg and L. Rodman, Minimal factorization of meromorphic matrix functions in terms of local data,Integral Equations and Operator Theory 10 (1987), 309–348.
[BR] 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.
[BGK] H. Bart, I. Gohberg and M.A. Kaashoek,Minimal Factorization of Matrix and Operator Functions, OT 1, Birkhäuser Verlag, Basel, 1979.
[B] V.A. Bolotnikov,Bitangential Caratheodory Problem in the Stieltjes Class, Kharkov University, 1990 (in Russian).
[D] J. Dieudonné,La Géometrie des Groupes Classiques, Springer-Verlag, Berlin, 1962.
[DK] Yu M. Dyukarev and V.E. Katsnel'son, Multiplicative and additive classes of Stieltjes analytic matrix-valued functions, and interpolation problems associated with them, I,Amer. Math. Soc. Transl. 131(2) (1986), 55–70.
[G] F.R. Gantmacher,The Theory of Matrices, Vols. I and II, Chelsea, New York, 1959.
[GK] I. Gohberg and M.A. Kaashoek, Regular rational matrix functions with prescribed pole and zero structure,Operator Theory: Advances and Applications 33 (I. Gohberg, ed.) (1988), 109–122.
[GKL] I. Gohberg, M.A. Kaashoek and P. Lancaster, Theory of regular matrix polynomials and band Toeplitz operators,Integral Equations and Operator Theory 11 (1988), 776–882.
[GKLR] I. Gohberg, M.A. Kaashoek, L. Lerer and L. Rodman, Minimal divisors of rational matrix functions with prescribed zero and pole structure,Operator Theory: Advances and Applications 12 (H. Dym and I. Gohberg, eds.) (1984), 241–275.
[GLR] I. Gohberg, P. Lancaster and L. Rodman,Invariant Subspaces of Matrices with Applications, J. Wiley and Sons, New York, 1986.
[KN] M.G. Krein and A.A. Nudelman, The Markov moment problem and extremal problems,Transl. Math. Monographs 50, Amer. Math. Soc., Providence, 1977.
[R] M. Rakowski, Generalized pseudoinverses of matrix valued functions,Integral Equations and Operator Theory 14 (1991), 564–585.
[RR] A.C.M. Ran and L. Rodman, Laurent interpolation for rational matrix functions and a local factorization principle,J. Math. Anal. Appl., to appear.
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Partially supported by NSF Grant 9000839 and by USA-Israel Binational Foundation Grant 88003094/2.
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Alpay, D., Ball, J.A., Gohberg, I. et al. State space theory of automorphisms of rational matrix functions. Integr equ oper theory 15, 349–377 (1992). https://doi.org/10.1007/BF01200324
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DOI: https://doi.org/10.1007/BF01200324