Abstract
The success of the theory of characteristic functions of nonselfadjoint operators and its applications to the System Theory [1–17] is the inspiration for attempts towards creating a general theory in the much more complicated case of several commuting nonselfadjoint operators. In this paper we study the close relations between sets of commuting operators in Hilbert space and related systems of partial differential equations. At the same time a generalization of the classical Cayley Hamilton Theorem, in the case of two commuting operators, is obtained which leads to unexpected connections with the theory of algebraic curves.
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References
Beurling, A. 1949 On two problems concerning linear transformation in Hilbert space, Acta Math. 81. 239–255.
De Branges, L. and Rovnjak, J. 1966 Canonical models in quantum scattering theory, Perturbation Theory and its Application in Quantum Mechanics (Pure. Adv. Sem. Math. Res. Center, U.S. Army, Theoret. Chem. Inst., Univ. of Wisconsin, Madison, Madison, Wisc. 1965) New York. Wiley, pp. 295–392.
Brodskii, M.S. 1971 Triangular and Jordan representations of linear operations; Transl. Math. Monogr., 32 Am. math. Soc., Providence, R.I.
Davis, C., Foias, C., Operators with bounded characteristic functions and their J-unitary dilation. Acta Sci. Math., 32.
Douglas, R.G., 1974 Canonical models, Math. Surveys v. 13.
Fuhrmann, P.A. 1) 1968 On the corona theorem and its application to spectral problems in Hilbert space, Trans. Amer. Math. Soc., 132, No. 1; 2) 1968 A functional calculus in Hilbert space based on operator valued analytic functions, Isr. J. Math., 6, No. 3.
Gohberg, I.C. and Krein, M.G. 1) 1969 Introduction to the theory of linear nonselfadjoint operators in Hilbert space, Transl. Math. Mon., v. 18; 2) 1970 Theory and applications of Volterra operators in Hilbert space, Transl. Math. Mon., v. 24.
Halmos, P.R. 1961 Shifts on Hilbert spaces. J. reine angew. Math. 208, 102–112.
Helson, H., 1979 Lectures on invariant subspaces, N.Y. Acad. Press.
Helton, J.W. 1976 Systems with infinite dimensional state space, Proc. I.E.E.E. v. 64, No. 1, Jan.
Howland, J. 1975 The Livsic matrix in perturbation theory, Journ. of Math. Anal. and Appl. 50, 415–437.
Lax, P.D. 1959 Translation invariant subspaces, Acta Math. 101, 163–173.
Lax, P.D., Phillips, R.G., 1969 Scattering Theory, Acad. Press, N.Y.
Livsic, M.S. 1973 Operators, Oscillations, Waves, Open Systems, Transl. Math. Mon., v. 34.
Livsic, M.S. and Jancevich, A.A. 1971 Theory of operator colligations in Hilbert space, Kharkov Univ.
Sarason, D. 1967 Generalized interpolation in H∞, Trans. Amer. Math. Soc., 127, No. 2, 179–203.
Sz.-Nagy, B., Foias, C., 1967 Analys harmonique des operators de l'espace de Hilbert, Akad. Kiado, Budapest.
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Livsic, M.S. Operator waves in Hilbert space and related partial differential equations. Integr equ oper theory 2, 25–47 (1979). https://doi.org/10.1007/BF01729359
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DOI: https://doi.org/10.1007/BF01729359