Bezout operators for analytic operator functions, I. A general concept of Bezout operator

[AJ] B.D.O. Anderson, E.I. Jury: Generalized Bezoutian and Sylvester matrices in multivariable linear control,IEEE Trans. Automat. Control AC-21 (1976), 551–556. [Ba] S. Barnett: A note on the Bezoutian matrix,SIAM J. Appl. Math. 22(1972), 84–86. [BGK] H. Bart, I. Gohberg, M. A. Kaashoek:Minimal Factorization of Matrix and Operator Functions, OT1. Birkhäuser, Basel, 1979. [B] E. Bezout: Recherches sur le degré des équations resultantes de l'évanoussement des inconnues, et sur les moyens qui'il convient d'employer pour trouver çes équations,Mém. Acad. Roy. Sci. Paris, 1794, 288–338. [BKAK] R.R. Bitmead, S.Y. Kung, B.D.O. Anderson, T. Kailath: Greatest common divisors via generalized Sylvester and Bezout matrices.IEEE Trans. Autom. Control AC-23(1978), 1043–1047. [CD] D. Carlson, B.N. Datta: On the effective computation of the inertia of a non-hermitian matrix,Numer. Math. 33(1979), 315–392. [C] A. Cayley: Note sur la méthode d'elimination de Bezout,J. Reine Angew. Math. 53(1857), 366–376. [CK] K.F. Clancey, B.A. Kon: The Bezoutian and the algebraic Riccati equation,Linear and Multilinear Algebra 15(1984), 265–278. [Da1] B.N. Datta: On the Routh-Hurwitz-Fujiwara and the Schur-Cohn-Fujiwara theorems for the root-separation problem,Lin. Algebra and Appl. 22(1978), 235–246. [Da2] B.N. Datta: Matrix equations, matrix polynomials and the number of zeros of a polynomial inside the unit circle,Linear and Multilinear Alg. 9(1980), 63–68. [D] H. Dym: A Hermite theorem for matrix polynomials.Operator Theory: Advances and Applications, (H. Bart, I. Gohberg, M.A. Kaashoek, eds.) 50(1991), 191–214. [DG] H. Dym, I. Gohberg: On an extension problem, generalized Fourier analysis and an entropy formula,Integral Equations and Operator Theory,3(1980), 143–215. [DY] H. Dym, N.Y. Young: A Schur-Cohen theorem for matrix polynomials,Proc. Edinburgh Math. Soc. 33(1990), 337–366. [F] M. Fujiwara: On algebraic equations whose roots lie in a circle or in a half-plane,Math. Z.,24(1926), 161–169 (in German). [Fu] P.A. Fuhrmann: On symmetric rational transfer functions,Linear Algebra and Appl. 50 (1983), 167–250. [GH] I. Gohberg, G. Heinig: The resultant matrix and its generalizations, II. Continual analog of resultant matrix,Acta Math. Acadm. Sci. Hungar.,28(1976), 198–209 (in Russian). [GLR] I. Gohberg, P. Lancaster, L. Rodman:Matrix Polynomials, Academic Press, New York, 1982. [GL] I. Gohberg, L. Lerer: Matrix generalizations of M.G. Krein theorems on matrix polynomials,Orthogonal Matrix-valued Polynomials and Applications (I. Gohberg, ed.). Operator Theory: Advances and Applications,34(1988) Birkhäuser, 137–202. [H] I. Haimovici:Operator Equations and Bezout Operators for Analytic Operator Functions, Ph.D. Thesis, Technion, Haifa, Israel, 1991 (in Hebrew). [HP] E. Hille, R.S. Phillips:Functional Analysis and Semi-groups, Amer. Math. Soc., Providence, RI, 1957. [HR] G. Heinig, K. Rost:Algebraic Methods for Toeplitz-like Matrices and Operators, Birkhäuser, Basel, 1984. [HF] U. Helmke, P.A. Fuhrmann: Bezoutians,Linear Algebra and Appl. 122-124 (1989), 1039–1097. [He] C. Hermite: Sur le nombre des racines d'une équation algébrique comprise entre des limites donnés,J. Reine Angew. Math. 52(1856), 39–51. [K] T. Kailath:Linear systems, Prentice-Hall, Englewood Cliffs, N.J., 1980. [Ka] R.E. Kalman: On the Hermite-Fujiwara theorem in stability theory,Quart. Appl. Math. 23(1965), 279–282. [Kr] N. Kravitsky: On the discriminant function of two commuting nonselfadjoint operators,Integral Equations and Operator Theory 3(1980), 97–125. [KN] M.G. Krein, M.A. Naimark: The method of symmetric and hermitian forms in theory of separation of the roots of algebraic equations, GNTI, Kharkov, 1936 (Russian);Linear and Multilinear Algebra 10(1981), 265–308. [LM] P. Lancaster, J. Maroulas: The kernel of the Bezoutian for operator polynomials,Linear and Multilinear Algebra 17(1985), 181–201. [LaT] P. Lancaster, M. Tismenetsky:The Theory of Matrices with Applications. Academic Press, New York, 1985. [La] F.L. Lander: The Bezoutian and inversion of Hankel and Toeplitz matrices,Mat. Issled. 9(1974), 49–57 (in Russian). [L] L. Lerer: The matrix quadratic equations and factorization of matrix polynomials,Operator theory: Advances and Applications 40(1989), 279–324. [LR] L. Lerer, L. Rodman: Spectral separation and inertia for operators polynomials,J. Math. Anal. and Appl. 169(1992), 260–282. [LRT] L. Lerer, L. Rodman, M. Tismenetsky: Bezoutian and the Schur-Cohn problem for operator polynomials,J. Math. Anal. Appl. 103(1984), 83–102. [LT1] L. Lerer, M. Tismenetsky: The eigenvalue separation problem for matrix polynomials,Integral Equations Operator Theory 5(1982), 386–445. [LT2] L. Lerer, M. Tismenetsky: Generalized Bezoutian and the inversion problem for block matrices, I. General scheme,Integral Equations Operator Theory 9(1986), 790–819. [LT3] L. Lerer, M. Tismenetsky: Toeplitz classification of matrices and inversion formulas, II. Block-Toeplitz and perturbed block-Toeplitz matrices, TR88.197, IBM-Israel Scientific Center, Haifa, Aug. 1986. [LT4] L. Lerer, M. Tismenetsky: Generalized Bezoutian and matrix equations,Linear Algebra and Appl. 99(1988), 123–160. [vdM] C.V.M. van der Mee: Realization and Linearization, Rapport nr. 109, Vrije Universiteit, Amsterdam, 1979. [M] B. Mitiagin: Linearization of holomorphic operator functions. I,II,Integral equations and operator theory,1(1978), 114–131, 226–249. [P] V. Ptak: Explicit expressions for Bezoutians,Linear Algebra and Appl. 59(1984), 43–54. [R] L. Rodman:An Introduction to Operator Polynomials, OT38, Birkhkauser Verlag, Basel, 1989. [S1] L.A. Sakhnovich: Operatorial Bezoutian in the theory of separation of roots of entire functions,Functional Analysis and its Applications 10(1976), 54–61 (in Russian). [S2] L.A. Sakhnovich: Factorization problems and operator identities,Uspekhi Mat. Nauk 41(1986), 1–55 (in Russian),Russian Math. Surveys 41(1986), 1–46. [Sy] I. Sylvester: On a theory of the syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm's functions, and that of the greatest algebraical common measure.Philos. Trans. Roy. Soc. London 143(1853), 407–548. [W] H.K. Wimmer: On the history of the Bezoutian and the resultant matrix,Linear Algebra and Appl. 128(1990), 27–34.